Revision sheet

Basic rules

• $(x^n)' = n x^{n-1}$; rewrite roots and fractions as powers first.
• Constants factor out; sums split; a constant alone gives 0.

Watch out: with negative exponents, $-2 - 1 = -3$.

Exponentials

• $(e^x)' = e^x$.
• $(b^x)' = b^x \ln b$.

Watch out: $x^n$ (variable base) and $b^x$ (variable exponent) have different rules.

Sine and cosine

• $(\sin x)' = \cos x$.
• $(\cos x)' = -\sin x$.

Watch out: the minus sign goes with the derivative of cosine; radians only.

Product rule

$(fg)' = f'g + fg'$: differentiate one factor at a time, keep the other.

Watch out: $(fg)' \ne f'g'$.

Quotient rule

$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$: "low d-high minus high d-low, over low squared".

Watch out: the order in the numerator matters.

tan, sec, cot, csc

• $(\tan x)' = \sec^2 x$, $(\sec x)' = \sec x\tan x$.
• $(\cot x)' = -\csc^2 x$, $(\csc x)' = -\csc x\cot x$.

Watch out: co-functions take a minus sign. You may also write $\sec^2 x$ as $\dfrac{1}{\cos^2 x}$.

Chain rule

$\big(f(g(x))\big)' = f'(g(x))\,g'(x)$: outside's derivative (inside kept) times inside's derivative.

Watch out: do not forget the factor $g'(x)$.

Inverse functions

• $(\ln x)' = \dfrac{1}{x}$, $(\ln u)' = \dfrac{u'}{u}$.
• $(\arcsin x)' = \dfrac{1}{\sqrt{1 - x^2}}$, $(\arctan x)' = \dfrac{1}{1 + x^2}$.

Watch out: the chain rule still applies to the inside.