4. Applications of Derivatives

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

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

In Exercises 51 and 52, find dp/dq.

p³ + 4pq - 3q² = 2

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

(x² + y²)² = (x – y)² at (1,0) and (1,–1)

58–59. Carry out the following steps.

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

xy^5/2+x^3/2y=12; (4, 1)

If x¹/³ + y¹/³ = 4, find d²y/dx² at the point (8, 8).

Use implicit differentiation to find dy/dx.

sin xy = x+y

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

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

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x(1−y²)+y³=0

The folium of Descartes (See Figure 3.27)

b. At what point other than the origin does the folium have a horizontal tangent line?

A challenging second derivative Find d²y/dx², where √y+xy=1.

Evaluate and simplify y'.

x = cos (x−y)

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

y = xe^y

In Exercises 27–32, find dy/dx.

ln y = e^y sinx

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

a. Verify that the given point lies on the curve.

(x²+y²)²=25/4 xy²; (1, 2)

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?

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