4. Applications of Derivatives

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

b. Graph the tangent lines on the given graph.

x+y³−y=1; x=1

13-26 Implicit differentiation Carry out the following steps.

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

tan xy = x+y; (0,0)

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

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

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.

xy + 2x - 5y = 2, (3, 2)

Differentiating Implicitly

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

x cos(2x + 3y) = y sin x

Evaluate and simplify y'.

xy⁴+x⁴y=1

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

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

13-26 Implicit differentiation Carry out the following steps.

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

³√x+³√y⁴ = 2;(1,1)

Vertical tangent lines

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

Given the equation $x{e}^y=x-y$, find $\frac{dy}{dx}$ using implicit differentiation.

Differentiating Implicitly

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

x²(x – y)² = x² – y²

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

b. Graph the tangent and normal lines on the given graph.

x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

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

x = y²

In Exercises 43–50, find by implicit differentiation.

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√xy = 1

Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>