The 8 derivatives to know by heart for the exam

The table to memorize

$f(x)$	$f'(x)$
$k$ (constant)	$0$
$x$	$1$
$x^n$	$n \cdot x^{n-1}$
$\dfrac{1}{x}$	$-\dfrac{1}{x^2}$
$\sqrt{x}$	$\dfrac{1}{2\sqrt{x}}$
$e^x$	$e^x$
$\ln(x)$	$\dfrac{1}{x}$
$\sin(x)$, $\cos(x)$	$\cos(x)$, $-\sin(x)$

The 3 assembly rules

• Sum: $(u + v)' = u' + v'$.
• Product: $(uv)' = u'v + uv'$.
• Quotient: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.

A small example

Differentiate $f(x) = 3x^4 + 2x - 5$:

• $(3x^4)' = 3 \cdot 4x^3 = 12x^3$
• $(2x)' = 2$
• $(-5)' = 0$
• Total: $f'(x) = 12x^3 + 2$.

The quotient mnemonic

« Bottom differentiates Top, minus Top differentiates Bottom, over Bottom squared » — order matters (sign changes).

The classic mistake

$(\ln(2x))' \ne \dfrac{1}{2x}$. You need the chain rule: $(\ln(2x))' = \dfrac{1}{2x} \times 2 = \dfrac{1}{x}$.

?Your turn

What is the derivative of f(x) = x⁵?

A.5x⁴B.x⁴C.5x⁶D.x⁵/5

Check