Chain, product and quotient rules

What you’ll learn

• Differentiate composite, product and quotient functions.
• Differentiate exponential, logarithmic and trigonometric functions.
• Find a reciprocal derivative when x is a function of y.

Understand the idea

The chain rule follows the layers of a composite function: differentiate the outer function while keeping the inner expression, then multiply by the derivative of the inner expression. This factor is essential for powers, logarithms, exponentials and trigonometric functions of anything other than plain $x$.

A product has two derivative contributions, one for the change of each factor. The quotient rule subtracts in a fixed order and squares the denominator. Algebraic simplification before differentiation may be easier than using the quotient rule directly. Trigonometric derivatives use angles measured in radians.

The reciprocal-function derivatives are $(\sec x)'=\sec x\tan x$, $(\csc x)'=-\csc x\cot x$ and $(\cot x)'=-\csc^2x$. Also, $(a^x)'=a^x\ln a$ for $a>0$. If $x$ is expressed as a function of $y$, calculate $dx/dy$ first and take its reciprocal to obtain $dy/dx$, where that derivative is non-zero. Continue to check the domain, especially for logarithms and denominators.

Key formulae

$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$

$$\frac{d}{dx}(uv)=u'v+uv',\qquad \frac{d}{dx}\left(\frac uv\right)=\frac{u'v-uv'}{v^2}$$

$$\frac{d}{dx}e^{kx}=ke^{kx},\qquad \frac{d}{dx}\ln x=\frac1x$$

$$\frac{d}{dx}\sin x=\cos x,\quad \frac{d}{dx}\cos x=-\sin x,\quad \frac{d}{dx}\tan x=\sec^2x$$

Worked example

Differentiate $y=x^2e^{3x}$.

1. Choose $u=x^2$ and $v=e^{3x}$.
2. Then $u'=2x$ and, by the chain rule, $v'=3e^{3x}$.
3. Apply the product rule: $y'=2xe^{3x}+3x^2e^{3x}$.

Answer: $y'=xe^{3x}(2+3x)$.