##
Differentiation Rules

Here's a handy summary of the differentiation rules you'll frequently use.

###
Product Rule

The differentiation rule for the product of two functions: $\begin{aligned}
(fg)'&= f'g + fg'\\[8px] &= [{\small \text{ (deriv of the 1st) }
\times \text{ (the 2nd) }}]\, + \,[{\small \text{ (the 1st) } \times \text{
(deriv of the 2nd)}}] \end{aligned}$
• For examples of the Product Rule,
visit our Calculating
Derivatives: Problems & Solutions page!

###
Quotient Rule

The differentiation rule for the quotient of two functions:
Many students remember the quotient rule by
thinking of the numerator as "hi," the demoninator as "lo," the derivative
as "d," and then singing

"lo d-hi minus hi d-lo over lo-lo"

• For examples of the Quotient Rule: Calculating
Derivatives: Problems & Solutions.

###
Chain Rule

The differentiation rule for the composition of two functions:
$\begin{aligned} \left[ f\Big(g(x)\Big)\right]' &= f'\Big(g(x)\Big) \cdot
g'(x) \\[8px] &= \text{[derivative of the outer function, evaluated at
the inner function]}\\[8px] & \qquad \times \text{[derivative of the inner
function]} \end{aligned}$ Alternatively, if we write $y = f(u)$ and $u =
g(x),$ then $\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$
Informally: $\dfrac{df}{dx} = \left[\dfrac{df}{d\text{(stuff)}}\text{,
with the same stuff inside} \right] \times \dfrac{d}{dx}\text{(stuff)}$ •
For *many* examples of the Chain Rule: Chain Rule: Problems &
Solutions.

One quick example: Consider $f(x) = (x^2 + 1)^7.$ To find the derivative,
think something like: "The function is a bunch of stuff to the 7th power.
So the derivative is 7 times that same stuff to the 6th power, times the
derivative of that stuff."
*Note:* You'd never actually write out "stuff = ...." Instead just
hold in your head what that "stuff" is, and proceed to write down the
required derivatives.

*Tip:* You can differentiate any function, for free, using Wolfram
*WolframAlpha's* Online Derivative
Calculator.