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. $f(x) = \dfrac{\sqrt{x}}{e^x\sin(x)}$

Trigonometric Functions

Recall the six trignometric functions and their relationship with each other. You can review them here.

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

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