3.4: The Product and Quotient Rules

We know how to take derivatives of sums, constants, and $x^n$. What about products?

Product Rule
\[\dfrac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)\]
Find the derivatives of the following:
  1. $f(x) = x^2\sin(x)$
  2. $f(x) = x^3(\sqrt{x} + 1)$

How about quotients?

Quotient Rule
\[\dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right] = \dfrac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}\]

You can remember this by thinking "low Dhigh minus high Dlow, square the bottom, away you go."

Differentiate (find the derivative) of the following:
  1. $f(x) = \dfrac{x}{2x-4}$
  2. $f(x) = \dfrac{x^2 + 1}{x^2-1}$
  3. $f(x) = \dfrac{\sqrt{x}}{e^x + 1}$

Trigonometric Functions

We can use the quotient rule to find derivatives of the trigonometric functions.

Differentiate the following:
  1. $f(x) = \tan(x)$
  2. $f(x) = \csc(x)$