Limit shortcuts

Standard limit (sin)

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

Foundation for all trig limits.

L'Hôpital's rule

$\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}$

Use when the limit is $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .

Differentiation rules

Power rule

$\frac{d}{dx}(x^n) = n x^{n-1}$

Works for any real exponent.

Product rule

$(fg)' = f'g + fg'$

Two functions multiplied — take turns differentiating each.

Quotient rule

$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

For ratios; remember the order $f'g$ before $fg'$ .

Chain rule

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

Outer-then-inner; the most common source of mistakes.

Common derivatives

sin

$\frac{d}{dx}\sin x = \cos x$

cos

$\frac{d}{dx}\cos x = -\sin x$

Note the negative sign.

e^x

$\frac{d}{dx} e^x = e^x$

The unique fixed-point function.

ln x

$\frac{d}{dx} \ln x = \frac{1}{x}$

Domain $x > 0$ .

Integral table

Power rule (integral)

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\quad(n \neq -1)$

Reverse of differentiation power rule.

1/x

$\int \frac{1}{x}\,dx = \ln|x| + C$

The $n=-1$ exception to the power rule.

sin / cos

$\int \sin x\,dx = -\cos x + C,\quad \int \cos x\,dx = \sin x + C$

Memorise the signs — easy to mix up.

Exponential

$\int e^x\,dx = e^x + C$

Same as its derivative.

Taylor / Maclaurin series

e^x

$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

Converges for all real $x$ .

sin x

$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Odd powers only.

cos x

$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$

Even powers only.

Calculus Formulas