4. Applications of Derivatives

Let f(x) = x².

a. Show that f(x)−f(y) / x−y = f′(x+y²), for all x≠y.

Implicit Differentiation

In Exercises 43–50, find by implicit differentiation.

xy + 2x + 3y = 1

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

y² = x² + 2x

Find $dy/dx$ for the equation below using implicit differentiation.

$5x^2+y^3=12$

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

sin y = 5x⁴−5; (1, π)

In Exercises 29 and 30, find the slope of the curve at the given points.

y² + x² = y⁴ – 2x at (–2,1) and (–2,–1)

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

sin y = 5x⁴−5; (1, π)

Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?

<IMAGE>

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.

sin x+sin y=y

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

a. Use implicit differentiation to find dy/dx.

Slopes, Tangent Lines, and Normal Lines

In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.

x²y² = 9, (–1,3)

In Exercises 43–50, find by implicit differentiation.

y² = x .

x + 1

45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

x⁴-x²y+y⁴=1; (−1, 1)

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

xy = cot(xy)