4. Applications of Derivatives

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)

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.

x²(3y²−2y³) = 4

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

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

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

Find by implicit differentiation.

x² + xy + y² - 5x = 2

Find by implicit differentiation.

x²y² = 1

In Exercises 51 and 52, find dp/dq.

q = (5p² + 2p)⁻³/²

Find the slope of the curve x³y³ + y² = x + y at the points (1, 1) and (1, -1).

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)

Implicit Differentiation

In Exercises 43–50, find by implicit differentiation.

xy + 2x + 3y = 1

In Exercises 43–50, find by implicit differentiation.

y² = x .

x + 1

In Exercises 51 and 52, find dp/dq.

p³ + 4pq - 3q² = 2

In Exercises 53 and 54, find dr/ds.

r cos 2s + sin²s = π

In Exercises 53 and 54, find dr/ds.

2rs - r - s + s² = -3