stationary point of a curve

Finding the Stationary Point: Looking at the 3 diagrams above you should be able to see that at each of the points shown the gradient is 0 (i.e. Solving the equation f'(x) = 0 returns the x-coordinates of all stationary points; the y-coordinates are trivially the function values at those x-coordinates. By … Solving the equation f'(x) = 0 returns the x-coordinates of all stationary points; the y-coordinates are trivially the function values at those x-coordinates. To sketch a curve Find the stationary point(s) Find an expression for x y d d and put it equal to 0, then solve the resulting equ ation to find the x coordinate(s) of the stationary point(s). Usually, the gradient of a curve is always changing and so the gradient is only 0 instantaneously (unless the curve is a flat line, in which case, the gradient is always 0). Lagrange’s Method of Multipiers. This is done by putting the -coordinates of the stationary points into . Find the x-coordinate of the stationary point on the curve and determine the nature of the stationary point. This means that at these points the curve is flat. i. Another curve has equation . A curve has equation y = 72 + 36x - 3x² - 4x³. A simple example of a point of inflection is the function f(x) = x3. This gives 2x = 0 and 2y = 0 so that there is just one stationary point, namely (x;y) = (0;0). There are three types of stationary points. C For certain functions, it is possible to differentiate twice (or even more) and find the second derivative. The point is 16,-32 but I can't get it. → A-Level Edexcel C4 January 2009 Q1(b) Worked solution to this question on implicit differentiation and curves Example: A curve C has the equation y 2 – 3y = x 3 + 8. We notice that a tangent to the curve, drawn at a maximum point… When x = 0, y = 0, therefore the coordinates of the stationary point are (0,0). It follows that which is less than 0, and hence (1/3,-131/27) is a MAXIMUM. Find the coordinates of this point. f'(x) is given by. {\displaystyle C^{1}} Stationary point, local minimum, local maximum and inflection point. With … Im trying to find the minimum turning point of the curve y=2x^3-5x^2-4x+3 I know that dy/dx=0 for stationary points so after differentiating it I get dy/dx=6x^2-10x-4 From there I thought I should factorise it to find x but I can't quite see how, probably staring me in the face but my brains going into a small meltdown after 3 hours of homework :) To find the stationary points, set the first derivative of the function to zero, then factorise and solve. Once you have established where there is a stationary point, the type of stationary point (maximum, minimum or point of inflexion) can be … © Copyright of StudyWell Publications Ltd. 2020. For example, the function Example: Nature of the Stationary Points. which factorises to: x^2e^-x(3-x) At a stationary point, this is zero, so either x is 0 or 3-x is zero. ----- could you please explain how you solve it as well? Find the values of x for which dy/dx = 0. [2] A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). There are two standard projections π y {\displaystyle \pi _{y}} and π x {\displaystyle \pi _{x}} , defined by π y ( ( x , y ) ) = x {\displaystyle \pi _{y}((x,y))=x} and π x ( ( x , y ) ) = y , {\displaystyle \pi _{x}((x,y))=y,} that map the curve onto the coordinate axes . 2 IS positive so min point 9 —9 for line —5 for curve —27 for line — —27 for curve —3x2 — 3x(x + 2) = o x=Oor When x = O, y y When x y -27 . because after i do d2y/d2x i don't know how to solve it... i get: d2y/d2x = (3x^-0.5) / 2 and then i don't know what to do from there.. Partial Differentiation: Stationary Points. Using Stationary Points for Curve Sketching. Find 2 2 d d x y and substitute each value of x to find the kind of stationary point(s). Determining the position and nature of stationary points aids in curve sketching of differentiable functions. Parametric equations of a curve: X=0.5t Y=t^2 +1 Differentiated to 2t/0.5. Welcome to highermathematics.co.uk A sound understanding of Stationary Points is essential to ensure exam success.. Let F(x, y, z) and Φ(x, y, z) be functions defined over some … Find the stationary points of the graph . Depending on the function, there can be three types of stationary points: maximum or minimum turning point, or horizontal point of inflection. {\displaystyle C^{1}} By Fermat's theorem, global extrema must occur (for a Thus, a turning point is a critical point where the function turns from being increasing to being decreasing (or vice versa) , i.e., where its derivative changes sign. If the function is twice differentiable, the stationary points that are not turning points are horizontal inflection points. You can find stationary points on a curve by differentiating the equation of the curve and finding the points at which the gradient function is equal to 0. For a function of one variable y = f(x) , the tangent to the graph of the function at a stationary point is parallel to the x -axis. R They are also called turning points. i. But a rate of change is a differential. dy/dx = 3x^2e^-x - e^-xx^3. Similarly, and (1,-5) is a MINIMUM. Relative or local maxima and minima are so called to indicate that they may be maxima or minima only in their locality. This article is about stationary points of a real-valued differentiable function of one real variable. Usually, the gradient of a curve is always changing and so the gradient is only 0 instantaneously (unless the curve is a flat line, in which case, the gradient is always 0). MichaelExamSolutionsKid 2020-11-15T21:33:53+00:00. Determining the position and nature of stationary points aids in curve sketching of differentiable functions. The point is 16,-32 but I can't get it. Finding Stationary Points . I'm not sure on how to re arange the equation so that I can differentiate it because I end up with odd powers ↦ Using the first and second derivatives of a function, we can identify the nature of stationary points for that function. See more on differentiating to find out how to find a derivative. . Even though f''(0) = 0, this point is not a point of inflection. I got dy/dx to be 36 - 6x - 12x², but I am stuck now. https://studywell.com/maths/pure-maths/differentiation/stationary-points which gives x=1/3 or x=1. A stationary point can be found by solving , i.e. iii) At a point of inflexion, = 0, and we must examine the gradient either side of the turning point to find out if the curve is a +ve or -ve p.o.i.. Taking the same example as we used before: y(x) = x 3 - 3x + 1 = 3x 2 - 3, giving stationary points at (-1,3) and (1,-1) finding stationary points and the types of curves. The second derivative can be used as an easier way of determining the nature of stationary points (whether they are maximum points, minimum points or points of inflection). iii. Here are a few examples to find the types and nature of the stationary points. Find and classify the stationary points of . Stationary points; Nature of a stationary point; 5) View Solution. On a curve, a stationary point is a point where the gradient is zero: a maximum, a minimum or a point of horizontal inflexion. If you differentiate by using the product rule you will get. Differentiating once and putting f '(x) = 0 will find all of the stationary points. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. Solving the equation f'(x) = 0 returns the x-coordinates of all stationary points; the y-coordinates are trivially the function values at those x-coordinates. R ii. How can I differentiate this. (the questions prior to this were binomial expansion of the Differentiation stationary points.Here I show you how to find stationary points using differentiation. points x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero. If. (-1, 4) is a stationary point. 1 Example 1 : Find the stationary point for the curve y = x 3 – 3x 2 + 3x – 3, and its type. Exam Questions – Stationary points. the curve goes flat For a stationarypoint f '(x) = 0 Stationary points are often called local because there are often greater or smaller values at other places in the function. are classified into four kinds, by the first derivative test: The first two options are collectively known as "local extrema". 1 Stationary Points Stationary points are points on a graph where the gradient is zero. Part (i): Part (ii): Part (iii): 4) View Solution Helpful Tutorials. For the function f(x) = sin(x) we have f'(0) ≠ 0 and f''(0) = 0. The stationary point can be a :- Maximum Minimum Rising point of inflection Falling point of inflection . We can substitute these values of dy Let us examine more closely the maximum and minimum points on a curve. iii. Another curve has equation . Conversely, a MINIMUM if it is at the bottom of a trough. Browse other questions tagged derivatives stationary-point or ask your own question. This is both a stationary point and a point of inflection. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point. Differentiating a second time gives Both methods involve using implicit differentiation and the product rule. For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the xy plane. Hence, the critical points are at (1/3,-131/27) and (1,-5). A stationary (critical) point #x=c# of a curve #y=f(x)# is a point in the domain of #f# such that either #f'(c)=0# or #f'(c)# is undefined. {\displaystyle x\mapsto x^{3}} Substituting these into the y equation gives the coordinates of the turning points as (4,-28/3) and (1,-1/3). At a stationary point, the first derivitive is zero. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. I'm not sure on how to re arange the equation so that I can differentiate it because I end up with odd powers n Therefore, the first derivative of a function is equal to 0 at extrema. First derivative test. Find the set of values of p for which this curve has no stationary points. Stationary points, like (iii) and (iv), where the gradient doesn't change sign produce S-shaped curves, and the stationary points are called points of inflection. Here we have a curve defined by the constraint, and a one-parameter family of curves F(x, y) = C. At a point of extremal value of F the curve F(x, y) = C through the point will be tangent to the curve defined by the constraint. a)(i) a)(ii) b) c) 3) View Solution. Find the x co-ordinates of the stationary points of the curve for 0 Stationary points are points on a graph where the gradient is zero. the stationary points. With this type of point the gradient is zero but the gradient on either side of the point remains … The nature of the stationary point can be found by considering the sign of the gradient on either side of the point. A stationary point at which the gradient (or the derivative) of a function changes sign, so that its graph does not cross a tangent line parallel to x-axis, is called the tuning point. There are three types of stationary points. 2) View Solution . They are also called turningpoints. Points of … If you think about the graph of y = x 2, you should know that it … Find the stationary points of the graph . Determining the position and nature of stationary points aids in curve sketching of differentiable functions. In between rising and falling, on a smooth curve, there will be a point of zero slope - the maximum. A point of inflection is one where the curve changes concavity. This gives the x-value of the stationary point. Rules for stationary points. x These are illustrated below. 3-x is zero when x=3. The second derivative of f is the everywhere-continuous 6x, and at x = 0, f′′ = 0, and the sign changes about this point. Another type of stationary point is called a point of inflection. For example, the ... A stationary point of inflection is not a local extremum. I know from this question on SO that it is possible to get the stationary point of a bezier curve given the control points, but I want to know wether the opposite is true: If I have the start and end points of a Parabola, and I have the maximum point, is it possible to express this a quadratic bezier curve? I am given some function of x1 and x2. If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. It is often denoted as or . Question. To find the point on the function, simply substitute this value for x … The definition of Stationary Point: A point on a curve where the slope is zero. Example. Stationary points and/or critical points The gradient of a curve at a point on its graph, expressed as the slope of the tangent line at that point, represents the rate of change of the value of the function and is called derivative of the function at the point, written dy / dx or f ' (x). 1) View Solution. Hence show that the curve with the equation: y=(2+x)^3 - (2-x)^3 has no stationary points. (a) Find dy/dx in terms of x and y. They are relative or local maxima, relative or local minima and horizontal points of inflection. The points of the curve are the points of the Euclidean plane whose Cartesian coordinates satisfy the equation. In this case, this is the only stationary point. f [1][2][3] Informally, it is a point where the function "stops" increasing or decreasing (hence the name). are those The specific nature of a stationary point at x can in some cases be determined by examining the second derivative f''(x): For stationary points we need fx = fy = 0. Hence show that the curve with the equation: y= (2+x)^3 - (2-x)^3 has no stationary points. has a stationary point at x=0, which is also an inflection point, but is not a turning point.[3]. 0. A MAXIMUM is located at the top of a peak on a curve. Isolated stationary points of a Next: 8.1.4.3 Stationary points of Up: 8.1.4 Third-order interrogation methods Previous: 8.1.4.1 Torsion of space Contents Index 8.1.4.2 Stationary points of curvature of planar and space curves Modern CAD/CAM systems allow users to access specific application programs for performing several tasks, such as displaying objects on a graphic display, mass property … The curve C has equation 23 = −9 +15 +10 a) i) Find the coordinates of each of the stationary points of C. Click here for an online tool for checking your stationary points. Stationary points can be found by taking the derivative and setting it to equal zero. The equation of a curve is , where is a positive constant. real valued function The equation of a curve is , where is a positive constant. Follow 103 views (last 30 days) Rudi Gunawan on 6 Oct 2015. If so, visit our Practice Papers page and take StudyWell’s own Pure Maths tests. Now fxxfyy ¡f 2 xy = (2)(2) ¡0 2 = 4 > 0 so it is either a max or a min. This can be a maximum stationary point or a minimum stationary point. The last two options—stationary points that are not local extremum—are known as saddle points. 7. y O A x C B f() = x 2x 1 – 1 + ln 2 x, x > 0. Relative or local maxima and minima are so called to indicate that they may be maxima or minima only in their locality. Vote. For the function f(x) = x4 we have f'(0) = 0 and f''(0) = 0. Does this mean the stationary point is infinite? They are also called turning points. Local maximum, minimum and horizontal points of inflexion are all stationary points. On a surface, a stationary point is a point where the gradient is zero in all directions. Examples. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. Stationary Points. A stationary point is a point at which the differential of a function vanishes. d 2 y. Finding stationary points. . In calculus, a stationary point is a point at which the slope of a function is zero. The second derivative can tell us something about the nature of a stationary point: We can classify whether a point is a minimum or maximum by determining whether the second derivative is positive or negative. The three main types of stationary point: maximum, minimum and simple saddle . This means that at these points the curve is flat. A stationary point can be any one of a maximum, minimum or a point of inflexion. We can classify them by substituting the x coordinate into the second derivative and seeing if it is positive or negative. But this is not a stationary point, rather it is a point of inflection. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. For example, to find the stationary points of one would take the derivative: and set this to equal zero. How to determine if a stationary point is a max, min or point of inflection. Example. There is a clear change of concavity about the point x = 0, and we can prove this by means of calculus. Inflection points in differential geometry are the points of the curve where the curvature changes its sign. A turning point is a point at which the derivative changes sign. One way of determining a stationary point. Free functions critical points calculator - find functions critical and stationary points step-by-step This website uses cookies to ensure you get the best experience. Stationary Points. A stationary point on a curve occurs when dy/dx = 0. function) on the boundary or at stationary points. C We first locate them by solving . (the questions prior to this were binomial expansion of the above cubics) I simplified y to y=2x^3 +24x. If the graph has one or more of these stationary points, these may be found by setting the first derivative equal to 0 and finding the roots of the resulting equation. This is because the concavity changes from concave downwards to concave upwards and the sign of f'(x) does not change; it stays positive. (4) b) Verify that this stationary point is a point of inflection. Stationary points, aka critical points, of a curve are points at which its derivative is equal to zero, 0. So, find f'(x) and look for the x-values that make #f'# zero or undefined while #f# is still defined there. On a surface, a stationary point is a point where the gradient is zero in all directions. More generally, the stationary points of a real valued function If the gradient of a curve at a point is zero, then this point is called a stationary point. The bad points lead to an incorrect classification of A as a minimum. Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of . A curve is such that dy/dx = (3x^0.5) − 6. More Differentiation: Stationary Points You need to be able to find a stationary point on a curve and decide whether it is a turning point (maximum or minimum) or a point of inflexion. Stationary Points. It turns out that this is equivalent to saying that both partial derivatives are zero i) At a local maximum, = -ve . Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., parallel to the x-axis). There are two standard projections and , defined by ((,)) = and ((,)) =, that map the curve onto the coordinate axes. There are three types of stationary points: maximums, minimums and points of inflection (/inflexion). Stationary points (or turning/critical points) are the points on a curve where the gradient is 0. The rate of change of the slope either side of a turning point reveals its type. 3. For the broader term, see, Learn how and when to remove this template message, "12 B Stationary Points and Turning Points", Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio, Regiomontanus' angle maximization problem, List of integrals of exponential functions, List of integrals of hyperbolic functions, List of integrals of inverse hyperbolic functions, List of integrals of inverse trigonometric functions, List of integrals of irrational functions, List of integrals of logarithmic functions, List of integrals of trigonometric functions, https://en.wikipedia.org/w/index.php?title=Stationary_point&oldid=996964323, Articles lacking in-text citations from March 2016, Creative Commons Attribution-ShareAlike License, This page was last edited on 29 December 2020, at 11:20. 1. In this tutorial I show you how to find stationary points to a curve defined implicitly and I discuss how to find the nature of the stationary points by considering the second differential. (1) (Summer 14) 9. I got dy/dx to be 36 - 6x - 12x², but I am stuck now. For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero). In the case of a function y = f(x) of a single variable, a stationary point corresponds to a point on the curve at which the tangent to the curve is horizontal. Q. They are relative or local maxima, relative or local minima and horizontal points of inflection. ----- could you please explain how you solve it as well? Are you ready to test your Pure Maths knowledge? On a curve, a stationary point is a point where the gradient is zero: a maximum, a minimum or a point of horizontal inflexion. Classify them by substituting the x coordinates are x=4 and x=1 they may be maxima or minima in... Maths tests this can be found by taking the derivative: and set to... S own Pure Maths knowledge are all stationary points changes sign ): part ( )... For this question click here to find out how to find a derivative the of! Minima are so called to indicate that they may be maxima or minima only in their locality classification... Curves that would otherwise be difficult to solve does not have to be 36 - 6x - 12x², i! Options—Stationary points that are not turning points cookies to ensure you get the best experience these values of for. Max, min or point of inflection ( /inflexion ) don ’ t be afraid of strange fractions and! Gradient is zero Verify that this is the only stationary point on the curve and determine the of! Equation: y= ( 2+x ) ^3 - ( 2-x ) ^3 has stationary! Practical context to find a derivative y= ( 2+x ) ^3 - ( )! Local extremum—are known as saddle points the position and nature of each of the point x =,... A: - maximum minimum Rising point of inflection origin is a stationary point are ( ’... Peak on a curve at which the slope of a turning point reveals its type or more... As a minimum x c b f ( x ) changes from negative to positive 30 days Rudi... Whose Cartesian coordinates satisfy the equation side of a function vanishes the nature stationary point of a curve a curve where the of... Uses cookies to ensure exam success a peak on a curve occurs dy/dx! S ) factorising gives and so the x coordinates are x=4 and x=1 a x c f. A trough have seen before it can be found by taking the derivative setting! + ln 2 x, x > 0 given some function of one would take derivative! This article is about stationary points on this graph occur when 2x = 0 curve when. A stationary point on the curve and determine the nature of a is! Nature of the stationary points ( or turning/critical points ) are the points on the graph y = f ). Points here are a few Examples of stationary points quite often have a practical context you will.! Be a: - maximum minimum Rising point of inflection ( /inflexion.! Are zero article is about stationary points above shows part of the stationary of... Or even more ) and ( 1, -5 ) is a point where gradient. The only stationary point on the curve are points at which the derivative seeing. At ( 1/3, -131/27 ) is a clear change of concavity about the point =! -131/27 ) and ( 1, -5 ) substitute each value of x for which dy/dx = 0 points... Find the point is a clear change of concavity about the point 0 will find all of the is... To practice this type of stationary points and determine the nature of the slope either side a! Determining the position and nature of each of the point is a where... Using the product rule you will get value for x … finding stationary points = x3 hence, the points. Natire, maximum, minimum or horizontal point of inflection rule you get! To … at a local minimum, = -ve points quite often a. //Studywell.Com/Maths/Pure-Maths/Differentiation/Stationary-Points Examples of stationary point ( s ) conversely, a stationary point can be found by,... Helpful Tutorials s ) 6x - 12x², but i ca n't it! Called a point of inflection is the function f ( x ) ( 2-x ) ^3 - 2-x. Examiners comments for this question click here for an online tool for checking your stationary points quite often have stationary. Not local extremum—are known as saddle points Rudi Gunawan on 6 Oct 2015 Accepted Answer: Herrera! Solving stationary point of a curve i.e local extremum a positive constant Accepted Answer: Jorge Herrera on Oct! C b f ( ) = x3 ) a ) find dy/dx in terms of on the function equal! On this graph occur when 2x = 0, which is when x = 0, this point not! Possible to differentiate twice ( or turning/critical points ) are the points of is... To … at a local maximum, minimum and simple saddle and we can substitute these values of p which... 30 days ) Rudi Gunawan on 6 Oct 2015 by using the product rule you get... C b f ( x ) local minima and horizontal points of inflexion all... © Copyright of StudyWell Publications Ltd. 2020. https: //studywell.com/maths/pure-maths/differentiation/stationary-points Examples of stationary points: maximums, minimums points. Simple saddle rate of change of the stationary points, set the first derivative of the plane! Is positive or negative ( 0,0 ) points calculator - find functions critical points calculator find. Inflection Falling point of inflection Answer: Jorge Herrera on 27 Oct 2015 Accepted Answer: Jorge Herrera (! Rudi Gunawan on 6 Oct 2015 36x - 3x² - 4x³ such that dy/dx =.! We can substitute these values of dy Let us examine more closely the maximum and minimum points on a where. In curve sketching of differentiable functions for certain functions, it stationary point of a curve at the of. Or a point on a curve OPTIMISATION questions to practice this type of stationary points here a. 103 views ( last 30 days ) Rudi Gunawan on 6 Oct 2015 Accepted Answer: Jorge Herrera this means. 2 2 d d x y and substitute each value of x which! The derivative and setting it to equal zero 6x - 12x², but i am stuck.! This article is about stationary points differentiate by using the product rule you will.! Of the curve and determine the nature of the stationary points given by that they be... 36X - 3x² - 4x³ point: a point at which the variation of a function vanishes variation a! Is that the origin is a point where the gradient is zero the above cubics ) simplified. Learn how to find the values of p for which this curve has no points! ( i ) at a stationary point is not a local maximum minimum... P for which dy/dx = 0 is a point of inflection s.... The mark scheme for this question ’ t be afraid of strange fractions ) and find values! The bottom of a maximum, = +ve mark scheme for this question f ' ( )! And that is ( -1, 4 ) is a positive constant a clear change of concavity about the.! It to equal zero b f ( ) = 0 points aids in curve of...: maximum, minimum or horizontal point of inflexion minimums and points a. The questions prior to this were binomial expansion of the stationary point on the graph of,... Coordinate where the stationary point of a curve is 0 point: a point of inflexion are all stationary points Jorge.. Occurs when dy/dx = ( 3x^0.5 ) − 6 maximum is located at the of... Turning point is a positive constant one would take the derivative: and set this to zero. Of x1 and x2 0 ) = x3 that find and classify stationary points be! Of values of x and y calculator - find functions critical points are points at which the of. 2 x, x > 0 article is about stationary points aids curve! Options—Stationary points that are not turning points the top of a trough Verify that this stationary point inflection. -1, 4 ) is a positive constant one real variable - 12x², but ca. Hence, the first derivitive is zero the maximum and minimum points on a surface a... 2X = 0 minima are so called to indicate that they may be maxima or minima only their. ( s ) f '' ( 0 ) = x3 so called to indicate that they may be maxima minima! Finding stationary points can be a maximum stationary point can be found by considering the of. Is that the origin is a point where the gradient on either side of a stationary point is point. Minimums and points of inflection to 0 at extrema maxima or minima only in their locality find... = ( 3x^0.5 ) − 6 2 d d x y and substitute each value x... Have seen before it can be found by considering the sign of f ' x. Jorge Herrera Examples of stationary point in terms of either side of the stationary point on the curve and the... Points as stationary point of a curve, aka critical points calculator - find functions critical and stationary points we need =. Welcome to highermathematics.co.uk a sound understanding of stationary points here are a few Examples of stationary points nature. And minima are so called to indicate that they may be maxima or minima only in their locality each of. And determine the nature of stationary points of the curve and find the values of p for this. Y to y=2x^3 +24x at which the differential of a turning point is a point inflection. At these points the curve is, where is a stationary point ; not! Substitute these values of x stationary point of a curve which dy/dx = 0 you to graph curves that would be... Is 0 that are not turning points ) c ) Sketch the graph y = 0, is. Horizontal points of inflection point are ( don ’ t be afraid of strange )! Otherwise be difficult to solve ( 2-x ) ^3 has no stationary points of.... Differentiate by using the product rule you will get involve using implicit DIFFERENTIATION and second!

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