Differentiation rules

Computing every derivative from first principles is tedious. Instead, we build a small library of rules and a table of standard derivatives. With these, the derivative of almost any algebraic-trigonometric expression follows by mechanical steps.

The five basic rules

Let $u(x), v(x)$ be differentiable functions, $c$ a constant.

(R1) Constant rule. $(c)' = 0$ .

(R2) Constant multiple. $(c \cdot u)' = c \cdot u'$ .

(R3) Sum/difference. $(u \pm v)' = u' \pm v'$ .

(R4) Product rule. $(u v)' = u' v + u v'$ .

(R5) Quotient rule. $\left(\dfrac{u}{v}\right)' = \dfrac{u' v - u v'}{v^2}$ , where $v \ne 0$ .

Proofs (sketch)

R4 proof: $(u(x+h)v(x+h) - u(x) v(x))/h$ . Add and subtract $u(x) v(x+h)$ : $= \dfrac{[u(x+h) - u(x)] v(x+h) + u(x)[v(x+h) - v(x)]}{h} = \dfrac{u(x+h) - u(x)}{h} v(x+h) + u(x) \dfrac{v(x+h) - v(x)}{h}.$ As $h \to 0$ : $u'(x) v(x) + u(x) v'(x)$ . $\qed$

R5 proof: similar idea, with more bookkeeping. Or derive R5 from R4 plus $(1/v)' = -v'/v^2$ .

Standard derivatives table

$f$	$f'$
$c$ (constant)	$0$
$x$	$1$
$x^n$	$n x^{n-1}$
$\sqrt{x}$	$1/(2\sqrt{x})$
$1/x$	$-1/x^2$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$1/x$
$\log_a x$	$1/(x \ln a)$

Worked examples

Example 1. $f(x) = 5 x^3 - 4 x + 7$ . Find $f'(x)$ .

By R2, R3: $f'(x) = 15 x^2 - 4$ .

Example 2. $f(x) = (3 x + 1)(2 x - 5)$ . Find $f'(x)$ .

By R4: $f'(x) = 3 \cdot (2 x - 5) + (3 x + 1) \cdot 2 = 6 x - 15 + 6 x + 2 = 12 x - 13$ .

(Or expand first: $f(x) = 6 x^2 - 13 x - 5 \Rightarrow f'(x) = 12 x - 13$ . ✓)

Example 3. $f(x) = \dfrac{x^2 + 1}{x - 1}$ . Find $f'(x)$ .

By R5: $f' = \dfrac{2 x \cdot (x - 1) - (x^2 + 1) \cdot 1}{(x - 1)^2} = \dfrac{2 x^2 - 2 x - x^2 - 1}{(x - 1)^2} = \dfrac{x^2 - 2 x - 1}{(x - 1)^2}$ .

Example 4. $f(x) = x \sin x$ . Find $f'(x)$ .

By R4: $f'(x) = 1 \cdot \sin x + x \cdot \cos x = \sin x + x \cos x$ .

Example 5. $f(x) = \dfrac{\sin x}{x}$ . Find $f'(x)$ .

By R5: $f'(x) = \dfrac{\cos x \cdot x - \sin x \cdot 1}{x^2} = \dfrac{x \cos x - \sin x}{x^2}$ .

Try it yourself

Find $f'(x)$ :

$f(x) = 7 x^4 - 3 x^2 + 2$ .
$f(x) = x^2 + 3 \sin x$ .
$f(x) = (x^2 + 1)(x - 2)$ .
$f(x) = \dfrac{x^3}{x + 1}$ .
$f(x) = \dfrac{1 + x}{1 - x}$ .
$f(x) = (x + 1)(x + 2)(x + 3)$ . (Use R4 twice or expand.)
$f(x) = x^2 \cos x$ .
$f(x) = \dfrac{\sin x}{1 + \cos x}$ .
$f(x) = \sqrt{x} \cdot \sin x$ .
$f(x) = e^x \cos x$ .
$f(x) = \dfrac{\ln x}{x}$ .
$f(x) = \dfrac{1}{1 + x^2}$ .

Pitfalls / Tricks

Product rule: $(u v)' \ne u' v'$ . Both terms must be present.
Quotient rule: numerator is $u' v - u v'$ (order matters).
For three-factor products, apply the product rule twice: $(u v w)' = u' v w + u v' w + u v w'$ .
Insight. Derivatives are linear: $(c u + d v)' = c u' + d v'$ . Whenever you see a linear combination, just differentiate term by term.