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Benginning Calculus Lecture notes 6 - implicit differentiation

This document discusses implicit differentiation and derivatives of inverse functions. It begins by introducing implicit differentiation and examples of taking derivatives of functions defined implicitly rather than explicitly. It then covers using implicit differentiation to find derivatives of inverse functions. Examples are provided such as finding the derivative of sin^-1(x) and tan^-1(x). The key concepts are using implicit differentiation to take derivatives where the relationship between variables is given implicitly rather than explicitly, and to take derivatives of inverse functions.

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Beginning Calculus - Implicit Di¤erentiation and Inverse Functions - Shahrizal Shamsuddin Norashiqin Mohd Idrus Department of Mathematics, FSMT - UPSI (LECTURE SLIDES SERIES) VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 1 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Leraning Outcomes Find the derivative of functions explicitly and Implicitly. Compute the derivatives of Inverse Functions VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 2 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Know: d dx (xn ) = nxn 1 , n 2 Z Don’t know: d dx xm/n , m n 2 Q, n 6= 0 VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 3 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Let y = xm/x . y = xm/n yn = xm d dx (yn ) = d dx (xm ) d dy (yn ) dy dx = mxm 1 , chain rule: dyn dx = dyn dy dy dx nyn 1 dy dx = mxm 1 dy dx = mxm 1 nyn 1 = m n xm 1 xm/n n 1 = axa 1 with a = m n VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 4 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Let x2 + y2 = 25. This is not a function. The equation implicitly de…nes y as several functions of x. -2 -1 1 2 -2 -1 1 2 x y x2 + y2 = 1 y = p 1 x2 VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 5 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Explicit Di¤erentiation Take the derivatives of y = p 1 x2 and y = p 1 x2 : d dx p 1 x2 = d dx 1 x2 1/2 = 1 2 1 x2 1/2 ( 2x) = x p 1 x2 d dx p 1 x2 = d dx 1 x2 1/2 = 1 2 1 x2 1/2 ( 2x) = x p 1 x2 VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 6 / 16
Implicit Di¤erentiation Derivative of Inverse Functions But sometimes it is not easy to di¤erentiate such equations, for example y4 + xy2 = 2 ) y2 = x p x2 4 ( 2) 2 ) y = s x p x2 + 8 2 for y de…ned explicitly as a function of x. VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 7 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Example d dx x2 + y2 = 1 : dy dx = x y = x p 1 x2 VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 8 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Example d dx x3 + y3 = 6xy : dy dx = x2 2y 2x y2 VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 9 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Example d dx y4 + xy2 = 2 : d dx y4 + xy2 = d dx 2 4y3 dy dx + y2 + 2xy dy dx = 0 dy dx = y2 4y3 + 2xy At x = 1, y = 1. So, d dx y4 + xy2 = 2 x=1 = 1 6 VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 10 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Example d dx sin (x + y) = y2 cos x : dy dx = cos (x + y) + y2 sin x 2y cos x cos (x + y) VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 11 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Example d2 dx2 x4 + y4 = 16 : d2y dx2 = 48x2 y7 First, …nd dy dx . Then d dx dy dx . VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 12 / 16
Implicit Di¤erentiation Derivative of Inverse Functions The Inverse Function Let y = p x, x > 0. Then, y2 = x. If we let f (x) = p x and g (y) = x, then g (y) = y2. -4 -2 2 4 -4 -2 2 4 g f VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 13 / 16
Implicit Di¤erentiation Derivative of Inverse Functions In general, If y = f (x) and g (y) = x, then g (f (x)) = x. g = f 1 and f = g 1. y = f (x) , f 1 (y) = x f f 1 (x) = x Implicit di¤erentiation allows us to …nd the derivative of any inverse function provided we know the derivative of the function. VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 14 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Example Let y = sin 1 x. d dx sin 1 x = 1 p 1 x2 Use: sin y = x. VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 15 / 16
Implicit Di¤erentiation Derivative of Inverse Functions Example Let y = tan 1 x. (Note: tan 1 x = arctan x). d dx tan 1 x = cos2 y = 1 x2 + 1 Use: tan y = x. VillaRINO DoMath, FSMT-UPSI (D5) Implicit Di¤erentiation and Inverse Functions 16 / 16