In Exercises 43–50, find by implicit differentiation.
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√xy = 1

Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

f(x) = x⁻¹, x₀ = 0.5, dx = 0.1

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = 3 csc(1 − 2√x)

Find the tangent line to the Witch of Agnesi (graphed here) at the point (2,1).

Find the derivatives of the functions in Exercises 1–42.

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𝔂 = ( √ x )²

( 1 + x )

Implicit Differentiation

In Exercises 43–50, find by implicit differentiation.

xy + 2x + 3y = 1

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

y² = x² + 2x