Derivatives in Differential Form


In Exercises 17–28, find dy.


y = 3 csc(1 − 2√x)

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

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

In Exercises 43–50, find by implicit differentiation.

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√xy = 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

In Exercises 65 and 66, find the derivative using the definition.

ƒ(t) = 1 .

2t + 1

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

y² + x² = y⁴ – 2x at (–2,1) and (–2,–1)