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Author: William A. Stein

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: (fg)=fg+fg=[ (deriv of the 1st) × (the 2nd) ]+[ (the 1st) × (deriv of the 2nd)]\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: [f(g(x))]=f(g(x))g(x)=[derivative of the outer function, evaluated at the inner function]×[derivative of the inner function]\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)y = f(u) and u=g(x),u = g(x), then dydx=dydududx\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} Informally: dfdx=[dfd(stuff), with the same stuff inside]×ddx(stuff)\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)=(x2+1)7.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.