4. Applications of Derivatives

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

In Exercises 43–50, find by implicit differentiation.

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√xy = 1

Differentiating Implicitly

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

x²(x – y)² = x² – y²

Differentiating Implicitly

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

xy = cot(xy)

Differentiating Implicitly

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

x⁴ + sin y = x³y²