Calculus: Early Transcendentals

James Stewart

Chapter 3

Differentiation Rules

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Section 5

Implicit Differentiation

Problem 1

(a) Find $ y' $ by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get $ y' $ in terms of $ x. $
(c) Check that your solutions to part (a) and (b) are consistent by substituting the expression for $ y $ into your solution for part (a).

$ 9 x^2 - y^2 = 1 $

Doruk I.

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Problem 2

(a) Find $ y' $ by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get $ y' $ in terms of $ x. $
(c) Check that your solutions to part (a) and (b) are consistent by substituting the expression for $ y $ into your solution for part (a).

$ 2x^2 + x + xy = 1 $

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Problem 3

(a) Find $ y' $ by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get $ y' $ in terms of $ x. $
(c) Check that your solutions to part (a) and (b) are consistent by substituting the expression for $ y $ into your solution for part (a).

$ \sqrt{x} + \sqrt{y} = 1 $

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Problem 4

(a) Find $ y' $ by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get $ y' $ in terms of $ x. $
(c) Check that your solutions to part (a) and (b) are consistent by substituting the expression for $ y $ into your solution for part (a).

$ \frac {2}{x} - \frac {1}{y} = 4 $

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Problem 5

Find $ dy/dx $ by implicit differentiation.

$ x^2 - 4xy + y^2 = 4 $

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Problem 6

Find $ dy/dx $ by implicit differentiation.

$ 2x^2 + xy - y^2 = 2 $

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Problem 6

Find $ dy/dx $ by implicit differentiation.

$ 2x^2 + xy - y^2 = 2 $

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Problem 7

Find $ dy/dx $ by implicit differentiation.

$ x^4 + x^2y^2 + y^3 = 5 $

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Problem 8

Find $ dy/dx $ by implicit differentiation.
$ x^3 - xy^2 + y^3 = 1 $

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Problem 9

Find $ dy/dx $ by implicit differentiation.
$ \frac {x^2}{x + y} = y^2 + 1 $

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Problem 10

Find $ dy/dx $ by implicit differentiation.
$ xe^y = x - y $

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Problem 11

Find $ dy/dx $ by implicit differentiation.
$ y \cos x = x^2 + y^2 $

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Problem 12

Find $ dy/dx $ by implicit differentiation.
$ \cos (xy) = 1 + \sin y $

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Problem 13

Find $ dy/dx $ by implicit differentiation.
$ \sqrt {x + y} = x^4 + y^4 $

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Problem 14

Find $ dy/dx $ by implicit differentiation.
$ e^y \sin x = x + xy $

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Problem 15

Find $ dy/dx $ by implicit differentiation.
$ e^{x/y} = x - y $

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Problem 16

Find $ dy/dx $ by implicit differentiation.
$ xy = \sqrt {x^2 + y^2} $

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Problem 17

Find $ dy/dx $ by implicit differentiation.
$ \tan^{-1} (x^2y) = x + xy^2 $

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Problem 18

Find $ dy/dx $ by implicit differentiation.
$ x \sin y + y \sin x = 1 $

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Problem 19

Find $ dy/dx $ by implicit differentiation.
$ \sin (xy) = \cos (x+y) $

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Problem 20

Find $ dy/dx $ by implicit differentiation.
$ tan (x - y) = \frac {y}{1 + x^2} $

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Problem 21

If $ f(x) + x^2[f(x)]^3 = 10 $ and $ f(1) = 2, $ find $ f'(1). $

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Problem 21

If $ f(x) + x^2[f(x)]^3 = 10 $ and $ f(1) = 2, $ find $ f'(1). $

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Problem 22

If $ g(x) + x \sin g(x) = x^2, $ find $ g'(0). $

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Problem 23

Regard $ y $ as the independent variable and $ x $ as the dependent variable and use implicit differentiation to find $ dx/dy. $
$ x^4y^2 - x^3y +2xy^3 = 0 $

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Problem 24

Regard $ y $ as the independent variable and $ x $ as the dependent variable and use implicit differentiation to find $ dx/dy. $
$ y \sec x = x \tan y $

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Problem 25

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ y \sin 2x = x \cos 2y, (\pi /2, \pi /4) $

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Problem 26

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ \sin (x + y) = 2x - 2y, (\pi, \pi) $

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Problem 27

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ x^2 - xy - y^2 = 1, (2, 1) $ (hyperbola)

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Problem 28

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ x^2 + 2xy + 4y^2 = 12, (2, 1) $ (ellipse)

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Problem 29

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ x^2 + y^2 = (2x^2 + 2y^2 - x)^2, (0, \frac {1}{2}), $ (cardiod)

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Problem 30

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ x^{\frac {2}{3}} + y^{\frac {2}{3}} = (-3 \sqrt{3}, 1), $ (astroid)

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Problem 31

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ 2(x^2 + y^2)^2 = 25(x^2 - y^2), (3, 1), $ (lemniscate)

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Problem 32

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
$ y^2 (y^2 - 4) = x^2 (x^2 - 5), (0, -2), $ (devil's curve).

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Problem 33

(a) The curve with equation $ y^2 = 5x^4 - x^2 $ is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2).
(b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing device will graph implicitly defined curves, then use that capability. If not. you can still graph this curve by graphing its upper and lower halves separately.)

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Problem 34

(a) The curve with equation $ y^2 = x^3 + 3x^2 $ is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, -2).
(b) At what points does this curve have horizontal tangents?
(c) Illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen.

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Problem 35

Find $ y" $ by implicit differentiation.
$ x^2 + 4y^2 = 4 $

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Problem 36

Find $ y" $ by implicit differentiation.
$ x^2 + xy + y^2 = 3 $

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Problem 37

Find $ y" $ by implicit differentiation.
$ \sin y + \cos x = 1 $

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Problem 38

Find $ y" $ by implicit differentiation.
$ x^3 - y^3 = 7 $

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Problem 39

If $ xy + e^y = e, $ find the value of $ y" $ at the point where $ x = 0. $

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Problem 40

If $ x^2 + xy + y^3 = 1, $ find the value of $ y" $ at the point where $ x = 1. $

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Problem 41

Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems,
(a) Graph the curve with equation
$ y(y^2 - 1)( y -2) = x(x -1)(x -2) $
At how many points does this curve have horizontal tangents? Estimate the $ x- $ coordinates of these points.
(b) Find equations of the tangent lines at the points (0, 1) and (0, 2).
(c) Find the exact r-coordinates of the points in part (a).
(d) Create even more fanciful curves by modifying the equation in part (a).

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Problem 42

(a) The curve with equation
$ 2y^3 + y^2 - y^5 = x^4 - 2x^3 + x^2 $
has been likened to a bouncing wagon. Use a computer algebra system to graph this curve and discover why.
(b) At how many points does this curve have horizontal tangent lines? Find the $ x- $ coordinates of these points.

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Problem 43

Find the points on the lemniscate in Exercise 31 where the tangent is horizontal.

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Problem 44

Show by implicit differentiation that the ellipse
$ \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1 $
at the point $ ( x_o, y_o) $ is
$ \frac {x_o x}{a^2} + \frac {y_o y}{b^2} = 1 $

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Problem 45

Find an equation of the tangent line to the hyperbola
$ \frac {x^2}{a^2} - \frac {y^2}{b^2} = 1 $
at the point $ (x_o, y_o). $

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Problem 46

Show that the sum of the $ x- $ and $ y- $ intercepts of any tangent line to the curve $ \sqrt x + \sqrt y = \sqrt c $ is equal to $ c. $

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Problem 47

Show, using implicit differentiation, that any tangent line at a point $ P $ to a circle $ O $ is perpendicular to the radius $ OP. $

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Problem 48

The Power Rule can be proved using implicit differentiation for the case where $ n $ in a rational number, $ n = p/q, $ and $ y = f(x) = x" $ is assumed beforehand to be a differentiable.
function. If $ y = x^{p/q}, $ then $ y^q = x^p. $ Use implicit differentiation to show that
$ y' = \frac {p}{q} x^{(p/q)-1} $

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Problem 49

Find the derivative of the function. Simplify where possible.
$ y = (tan^{-1} x)^2 $

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Problem 50

Find the derivative of the function. Simplify where possible.
$ y = \tan^{-1} (x^2) $

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Problem 51

Find the derivative of the function. Simplify where possible.
$ y = \sin^1 (2x + 1) $

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Problem 52

Find the derivative of the function. Simplify where possible.
$ g(x) = \arccos \sqrt x $

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Problem 53

Find the derivative of the function. Simplify where possible.
$ F(x) = x \sec^{-1} (x^3) $

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Problem 54

Find the derivative of the function. Simplify where possible.
$ y = \tan{-1} (x - \sqrt {1 + x^2}) $

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Problem 55

Find the derivative of the function. Simplify where possible.
$ h(t) = \cot^{-1} (t) + \cot{-1} (1/t) $

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Problem 56

Find the derivative of the function. Simplify where possible.
$ R(t) = \arcsin (1/t) $

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Problem 57

Find the derivative of the function. Simplify where possible.
$ y = x \sin^{-1} x + \sqrt{1 -x^2} $

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Problem 58

Find the derivative of the function. Simplify where possible.
$ y = \cos^{-1} (\sin^{-1} t) $

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Problem 59

Find the derivative of the function. Simplify where possible.
$ y = \arccos ( \frac {b + a \cos x}{a + b \cos x}), 0 \le x \le \pi, a > b > 0 $

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Problem 60

Find the derivative of the function. Simplify where possible.
$ y = \arctan \sqrt {\frac {1 - x}{1 + x}} $

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Problem 61

Find $ f'(x). $ Check that your answer is reasonable by comparing the graphs of $ f $ and $ f'. $
$ f(x) = \sqrt {1 - x^2} \arcsin x $

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Problem 62

Find $ f'(x). $ Check that your answer is reasonable by comparing the graphs of $ f $ and $ f'. $
$ f(x) = \arctan (x^2 - x) $

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Problem 63

Prove the formula for $ (d/dx) (\cos^{-1} x) $ by the same method as for $ (d/dx) (\sin^{-1} x). $

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Problem 64

(a) One way of defining $ \sec^{-1} x $ is to say that $ y= \sec^{-1} x \Leftrightarrow \sec y = x $ and $ 0 \le y < \pi/2 $ or $ \pi \le y < 3\pi/2. $ Show that, with this definition,
$ \frac {d}{dx} (\sec^{-1} x) = \frac {1}{x \sqrt {x^2 - 1}} $
(b) Another way of defining $ \sec^{-1} x \Leftrightarrow \sec y = x $ and $ 0 \le y \le \pi, y \not= \pi/2. $ Show that, with this definition,
$ \frac {d}{dx} (\sec{-1} x) = \frac {1}{\mid x \mid \sqrt {x^2 -1}} $

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Problem 65

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.
$ x^2 + y^2 = r^2, ax + by = 0 $

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Problem 66

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.
$ x^2 + y^2 = ax, x^2 + y^2 = by $

Frank L.

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Problem 67

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.
$ y = cx^2, x^2 + 2y^2 = k $

Chris T.

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Problem 68

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.
$ y = ax^3, x^2 + 3y^2 = b $

Chris T.

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Problem 69

Show that the ellipse $ x^2/a^2 + y^2/b^2 = 1 $ and the hyperbola $ x^2/A^2 - y^2/B^2 = 1 $ are orthogonal trajectories if $ A^2 < a^2 $ and $ a^2 - b^2 = A^2 + B^2 $ (so the ellipse and hyperbola have the same foci)

Suman Saurav T.

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Problem 70

Find the value of the number $ a $ such that the families of curves $ y = (x +c)^{-1} $ and $ y = a(x + k)^{1/3} $ are orthogonal trajectories.

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Problem 71

(a) The van van der Waals equation for $ n $ moles of a gas is
$ (P + \frac {n^2 a}{V^2})(V - nb) = nRT $
where $ P $ is the pressure, $ V $ is the volume, and $ T $ is the temperature of the gas. The constant $ R $ is the universal gas constant and $ a $ and $ b $ are positive constants that are characteristic of a particular gas. If $ T $ remains constant, use implicit differentiation to rind $ dV/dP. $
(b) Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of $ V = 10L $ and a pressure of $ P = 2.5 $ atm. Use $ a = 3.592 L^2 -atm/mole^2 $ and $ b = 0.04267 L/mole, $

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Problem 72

(a) Use implicit differentiation to find $ y' $ if
$ x^2 + xy + y^2 + 1 = 0 $
(b) Plot the curve in part (a). What do you see? Prove that what you see is correct.
(c) In view of part (b), what can you say about the expression for $ y' $ that you found in part (a)?

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Problem 73

The equation $ x^2 - xy + y^2 = 3 $ represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinates axes. Find the points at which this ellipse crosses the $ a-axis $ and show that the tangent lines at these points are parallel.

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Problem 74

(a) Where does the normal line to the ellipse $ x^2 - xy + y^2 = 3 $ at the point (-1, 1) intersect the ellipse a second time?
(b) Illustrate part (a) by graphing the ellipse and the normal line.

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Problem 75

Find all points on the curve $ x^2 y^2 + xy = 2 $ where the slope of the tangent line is -1.

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Problem 76

Find equations of both the tangent lines to the ellipse $ x^2 + 4y^2 = 36 $ that pass through the point (12, 3).

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Problem 77

(a) Suppose $ f $ is a one-to-one differentiable function and its inverse function $ f^{-1} $ is also differentiable. Use implicit differentiation to show that
$ (f^{-1})'(x) = \frac {1}{f'(f^{-1}(x))} $
provided that the denominator is not 0.
(b) If $ f(4) = 5 $ and $ f'(4) = \frac {2}{3}, $ find $ (f^{-1})'(5). $

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Problem 78

(a) Show that $ f(x) = x + r^x $ is one-to-one.
(b) What is the value of $ f^{-1}(1)? $
(c) Use the formula from Exercise 77(a) to find $ (f^{-1})'(1). $

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Problem 79

The Bessel function of order 0, $ y = J (x), $ satisfies the differential equation $ xy" + y' + xy = 0 $ for all values of $ x $ and its value at 0 is $ J(0) = 1. $
(a) Find $ J'(0). $
(b) Use implicit differentiation to find $ J'(0). $

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Problem 80

The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region $ x^2 + 4y^2 \le 5. $ If the point (-5, 0) is on the edge of the shadow, how far above far above the x-axis is the lamp located?

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