4. Applications of Derivatives

5–8. Calculate dy/dx using implicit differentiation.

e^y-e^x = C, where C is constant

Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

x⁴+y⁴ = 2;(1,−1)

13-26 Implicit differentiation Carry out the following steps.

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

x⁴+y⁴ = 2;(1,−1)

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

x = e^y; (2, ln 2)

13-26 Implicit differentiation Carry out the following steps.

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

x = e^y; (2, ln 2)

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

sin y = 5x⁴−5; (1, π)

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)

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

sin x+sin y=y

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

y = xe^y

Vertical tangent lines

b. Does the curve have any horizontal tangent lines? Explain.

Vertical tangent lines

b. Does the curve have any horizontal tangent lines? Explain.

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)