## Differentiation Rules

## 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{align*}

(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{align*}

### Quotient Rule

The differentiation rule for the quotient of two functions:

\begin{align*}

\dfrac{d}{dx}\left(\dfrac{f}{g} \right) &= \dfrac{\left(\dfrac{d}{dx}f \right)g – f\left(\dfrac{d}{dx}g \right)}{g^2} \\[8px]

&=\dfrac{{[{\small \text{(deriv of numerator) } \times \text{ (denominator)}]}\\ \quad – \, [{\small \text{ (numerator) } \times \text{ (deriv of denominator)}}]}}{{\small \text{all divided by [the denominator, squared]}}}

\end{align*}

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”

### Chain Rule

The differentiation rule for the composition of two functions:

\begin{align*}

\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{align*}

Alternatively, if we write $y = f(u)$ and $u = g(x),$ then

$$\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} $$

$$\dfrac{df}{dx} = \left[\dfrac{df}{d\text{(stuff)}}\text{, with the same stuff inside} \right] \times \dfrac{d}{dx}\text{(stuff)}$$

• For

*many*examples of the Chain Rule: Chain Rule: Problems & Solutions.

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.”

\begin{align*}

f(x) &= (\text{stuff})^7; \quad \text{stuff} = x^2 + 1 \\[12px]

\text{Then}\phantom{f(x)= }\\

\frac{df}{dx} &= 7(\text{stuff})^6 \cdot \left(\frac{d}{dx}(x^2 + 1)\right) \\[8px]

&= 7(x^2 + 1)^6 \cdot (2x) \quad \cmark

\end{align*}

*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.