Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

[Technology Exercise]

Graph the curves in Exercises 39–48.

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.

y = 4x²/⁵ − 2x

In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?

54. x³ + y² = sin²y

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

y = x² + x + 8

Finding Linearizations

In Exercises 1–5, find the linearization L(x) of f(x) at x = a.

f(x) = ∛x, a = −8

Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².