Generalizing the Product Rule The Derivative Product Rule gives the formula


d/dx (uv) = u (dv/dx) + (du/dx) v


for the derivative of the product uv of two differentiable functions of x.


b. What is the formula for the derivative of the product u₁u₂u₃u₄ of four differentiable functions of x?

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

𝔂 = 1 x² csc 2

2 x

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

𝔂 = x⁻¹/² sec (2x)²

Find all points (x, y) on the graph of y = x/(x − 2) with tangent lines perpendicular to the line y = 2x + 3.

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

Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,

d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹

where m is a positive integer.

Assume that functions f and g are differentiable with f(1) = 2, f'(1) = −3, g(1) = 4, and g'(1) = −2. Find the equation of the line tangent to the graph of F(x) = f(x)g(x) at x = 1.

Assume that functions f and g are differentiable with f(2) = 3, f'(2) = −1, g(2) = −4, and g'(2) = 1. Find an equation of the line perpendicular to the line tangent to the graph of F(x) = (f(x) + 3) / (x − g(x)) at x = 2.

Find the derivatives of the functions in Exercises 17–28.

y = ((x + 1)(x + 2)) / ((x − 1)(x − 2))