4. Applications of Derivatives

Vertical tangent lines

a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).

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)

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

$3\sqrt{y}=x-y$

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

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

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

$xy=\sin\left(y\right)$

66–71. Higher-order derivatives Find and simplify y''.

x + sin y = y

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

x = y²

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

sin y+2 = x

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

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

Use implicit differentiation to find dy/dx.

sin xy = x+y

Use implicit differentiation to find dy/dx.

e^xy = 2y

Use implicit differentiation to find dy/dx.

cos y² + x = e^y

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

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

6x³+7y³ = 13xy