4. Applications of Derivatives

The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y⁴ – 4y² = x⁴ – 9x² at the four indicated points.

In Exercises 51 and 52, find dp/dq.

q = (5p² + 2p)⁻³/²

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

³√x+³√y⁴ = 2;(1,1)

Find by implicit differentiation.

x²y² = 1

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

Use implicit differentiation to find dy/dx.

e^xy = 2y

In Exercises 53 and 54, find dr/ds.

r cos 2s + sin²s = π

In Exercises 27–32, find dy/dx.

3+siny = y-x^3

Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².

b. Evaluate this derivative when r=30 and h=40.

51–56. Second derivatives Find d²y/dx².

x⁴+y⁴ = 64

Parallel tangent lines Find the two points where the curve x² + xy + y² = 7 crosses the x-axis, and show that the tangent lines to the curve at these points are parallel. What is the common slope of these tangent lines?

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

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x + √xy = 6, (4, 1)

13-26 Implicit differentiation Carry out the following steps.

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

(x+y)^2/3=y; (4, 4)

In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?

54. x³ + y² = sin²y

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

6x³+7y³ = 13xy