The figure shows a lamp located three units to the right of the y-axis and a shadow

(a) Find y9 by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y9 in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 9x 2 2 y 2 1

(a) Find y9 by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y9 in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 2x 2 1 x 1 xy 1

(a) Find y9 by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y9 in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). sx 1 sy 1

(a) Find y9 by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y9 in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 2 x 2 1 y 4

Find dyydx by implicit differentiation. x 2 2 4xy 1 y 2 4

Find dyydx by implicit differentiation. 2x 2 1 xy 2 y 2 2

Find dyydx by implicit differentiation. x 4 1 x 2 y 2 1 y 3 5

Find dyydx by implicit differentiation. x 3 2 xy 2 1 y 3 1

Find dyydx by implicit differentiation. x 2 x 1 y y 2 1 1

Find dyydx by implicit differentiation. xey x 2 y

Find dyydx by implicit differentiation. y cos x x 2 1 y 2

Find dyydx by implicit differentiation. cossxyd 1 1 sin y

Find dyydx by implicit differentiation. sx 1 y x 4 1 y 4

Find dyydx by implicit differentiation. ey sin x x 1 xy

Find dyydx by implicit differentiation. exyy x 2 y

Find dyydx by implicit differentiation.xy sx 2 1 y 2

Find dyydx by implicit differentiation. tan21 sx 2 yd x 1 xy 2

Find dyydx by implicit differentiation.x sin y 1 y sin x 1

Find dyydx by implicit differentiation. sinsxyd cossx 1 yd

Find dyydx by implicit differentiation.tansx 2 yd y 1 1 x 2

If f sxd 1 x2 f fsxdg3 10 and fs1d 2, find f9s1d

If tsxd 1 x sin tsxd x 2 , find t9s0d.

Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dxydy. x 4 y2 2 x 3 y 1 2xy3 0

Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dxydy. y sec x x tan y

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin 2x x cos 2y, sy2, y4d

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. sinsx 1 yd 2x 2 2y, s, d

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x 2 2 xy 2 y 2 1, s2, 1d (hyperbola)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x 2 1 2xy 1 4y 2 12, s2, 1d (ellipse)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 1 y2 s2x2 1 2y2 2 xd 2 , s0, 1 2d, (cardioid)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x 2y3 1 y 2y3 4, s23s3, 1d, (astroid)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 2sx 2 1 y 2 d 2 25sx 2 2 y 2 d, (3, 1), (lemniscate)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.y2 s y2 2 4d x2 sx2 2 5d, (0, 22), (devils curve)

(a) The curve with equation y 2 5x 4 2 x 2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point s1, 2d. (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.)

(a) The curve with equation y 2 x 3 1 3x 2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point s1, 22d. (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.

Find y99 by implicit differentiation. x 2 1 4y 2 4

Find y99 by implicit differentiation. x 2 1 xy 1 y 2 3

Find y99 by implicit differentiation. sin y 1 cos x 1

Find y99 by implicit differentiation. x 3 2 y 3 7

If xy 1 ey e, find the value of y99 at the point where x 0.

If x 2 1 xy 1 y 3 1, find the value of y999 at the point where x 1

Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. (a) Graph the curve with equation ys y 2 2 1ds y 2 2d xsx 2 1dsx 2 2d 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 x-coordinates of the points in part (a). (d) Create even more fanciful curves by modifying the equation in part (a).

(a) The curve with equation 2y 3 1 y 2 2 y 5 x 4 2 2x 3 1 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.

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

Show by implicit differentiation that the tangent to the ellipse x 2 a2 1 y 2 b2 1 at the point sx0, y0 d is x0 x a2 1 y0 y b2 1

Find an equation of the tangent line to the hyperbola x 2 a2 2 y 2 b2 1 at the point sx0, y0d.

Show that the sum of the x- and y-intercepts of any tangent line to the curve sx 1 sy sc is equal to c.

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

The Power Rule can be proved using implicit differentiation for the case where n is a rational number, n pyq, and y fsxd x n is assumed beforehand to be a differentiable function. If y x pyq , then y q x p . Use implicit differentiation to show that y9 p q xs pyqd21

Find the derivative of the function. Simplify where possible. y stan21 xd 2

Find the derivative of the function. Simplify where possible. y tan21 sx 2 d

Find the derivative of the function. Simplify where possible. y sin21 s2x 1 1d

Find the derivative of the function. Simplify where possible. tsxd arccossx

Find the derivative of the function. Simplify where possible.Fsxd x sec21 sx 3 d

Find the derivative of the function. Simplify where possible. y tan21 sx 2 s1 1 x 2 d

Find the derivative of the function. Simplify where possible. hstd cot21 std 1 cot21 s1ytd

Find the derivative of the function. Simplify where possible.Rstd arcsins1ytd

Find the derivative of the function. Simplify where possible.y x sin21 x 1 s1 2 x 2

Find the derivative of the function. Simplify where possible. y cos21 ssin21 td

Find the derivative of the function. Simplify where possible. y arccosS b 1 a cos x a 1 b cos x D, 0 < x < , a . b . 0

Find the derivative of the function. Simplify where possible.y arctan 1 2 x 1 1 x

Find f9sxd. Check that your answer is reasonable by comparing the graphs of f and f9. fsxd s1 2 x 2 arcsin x

Find f9sxd. Check that your answer is reasonable by comparing the graphs of f and f9.fsxd arctansx 2 2 x

Prove the formula for sdydxdscos21 xd by the same method as for sdydxdssin21 xd.

(a) One way of defining sec21 x is to say that y sec21 x &? sec y x and 0 < y , y2 or  < y , 3y2. Show that, with this definition, d dx ssec21 xd 1 xsx 2 2 1 (b) Another way of defining sec21 x that is sometimes used is to say that y sec21 x &? sec y x and 0 < y < , y y2. Show that, with this definition, d dx ssec21 xd 1 | x |sx 2 2 1

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 1 y 2 r 2 , ax 1 by 0

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 1 y 2 ax, x 2 1 y 2 by

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 1 2y 2 k

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 1 3y 2 b

Show that the ellipse x 2 ya2 1 y 2 yb2 1 and the hyperbola ; x 2 yA2 2 y 2 yB2 1 are orthogonal trajectories if A2 , a2 and a2 2 b2 A2 1 B2 (so the ellipse and hyperbola have the same foci).

Find the value of the number a such that the families of curves y sx 1 cd 21 and y asx 1 kd 1y3 are orthogonal trajectories.

(a) The van der Waals equation for n moles of a gas is SP 1 n2 a V 2 DsV 2 nbd 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 find dVydP. (b) Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of V 10 L and a pressure of P 2.5 atm. Use a 3.592 L2 -atmymole2 and b 0.04267 Lymole.

(a) Use implicit differentiation to find y9 if x 2 1 xy 1 y 2 1 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 y9 that you found in part (a)?

The equation x 2 2 xy 1 y 2 3 represents a rotated ellipse, that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.

(a) Where does the normal line to the ellipse x 2 2 xy 1 y 2 3 at the point s21, 1d intersect the ellipse a second time? (b) Illustrate part (a) by graphing the ellipse and the normal line.

Find all points on the curve x 2 y 2 1 xy 2 where the slope of the tangent line is 21.

Find equations of both the tangent lines to the ellipse x 2 1 4y 2 36 that pass through the point s12, 3d.

(a) Suppose f is a one-to-one differentiable function and its inverse function f 21 is also differentiable. Use implicit differentiation to show that s f 21 d9sxd 1 f9s f 21 sxdd provided that the denominator is not 0. (b) If fs4d 5 and f9s4d 2 3, find s f 21 d9s5d.

(a) Show that fsxd x 1 ex is one-to-one. (b) What is the value of f 21 s1d? (c) Use the formula from Exercise 77(a) to find s f 21 d9s1d.

The Bessel function of order 0, y Jsxd, satisfies the differential equation xy99 1 y9 1 xy 0 for all values of x and its value at 0 is Js0d 1. (a) Find J9s0d. (b) Use implicit differentiation to find J99s0d

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 1 4y 2 < 5. If the point s25, 0d is on the edge of the shadow, how far above the x-axis is the lamp located?