Key Equations

Difference quotient	$Q = \frac{f (x) - f (a)}{x - a}$
Difference quotient with increment $h$	$Q = \frac{f (a + h) - f (a)}{a + h - a} = \frac{f (a + h) - f (a)}{h}$
Slope of tangent line	$m_{tan} = lim_{x \to a} \frac{f (x) - f (a)}{x - a}$ $m_{tan} = lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$
Derivative of $f (x)$ at $a$	$f^{'} (a) = lim_{x \to a} \frac{f (x) - f (a)}{x - a}$ $f^{'} (a) = lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$
Average velocity	$v_{a ve} = \frac{s (t) - s (a)}{t - a}$
Instantaneous velocity	$v (a) = s^{'} (a) = lim_{t \to a} \frac{s (t) - s (a)}{t - a}$

The derivative function

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

Derivative of sine function	$\frac{d}{d x} (sin x) = cos x$
Derivative of cosine function	$\frac{d}{d x} (cos x) = − sin x$
Derivative of tangent function	$\frac{d}{d x} (tan x) = {sec}^{2} x$
Derivative of cotangent function	$\frac{d}{d x} (cot x) = − {csc}^{2} x$
Derivative of secant function	$\frac{d}{d x} (sec x) = sec x tan x$
Derivative of cosecant function	$\frac{d}{d x} (csc x) = − csc x cot x$

The chain rule	$h^{'} (x) = f^{'} (g (x)) g^{'} (x)$
The power rule for functions	$h^{'} (x) = n {(g (x))}^{n - 1} g^{'} (x)$

Inverse function theorem	${(f^{−1})}^{'} (x) = \frac{1}{f^{'} (f^{−1} (x))}$ whenever $f^{'} (f^{−1} (x)) \neq 0$ and $f (x)$ is differentiable.
Power rule with rational exponents	$\frac{d}{d x} (x^{m / n}) = \frac{m}{n} x^{(m / n) - 1} .$
Derivative of inverse sine function	$\frac{d}{d x} {sin}^{−1} x = \frac{1}{\sqrt{1 - {(x)}^{2}}}$
Derivative of inverse cosine function	$\frac{d}{d x} {cos}^{−1} x = \frac{−1}{\sqrt{1 - {(x)}^{2}}}$
Derivative of inverse tangent function	$\frac{d}{d x} {tan}^{−1} x = \frac{1}{1 + {(x)}^{2}}$
Derivative of inverse cotangent function	$\frac{d}{d x} {cot}^{−1} x = \frac{−1}{1 + {(x)}^{2}}$
Derivative of inverse secant function	$\frac{d}{d x} {sec}^{−1} x = \frac{1}{\| x \| \sqrt{{(x)}^{2} - 1}}$
Derivative of inverse cosecant function	$\frac{d}{d x} {csc}^{−1} x = \frac{−1}{\| x \| \sqrt{{(x)}^{2} - 1}}$

Derivative of the natural exponential function	$\frac{d}{d x} (e^{g (x)}) = e^{g (x)} g^{'} (x)$
Derivative of the natural logarithmic function	$\frac{d}{d x} (ln g (x)) = \frac{1}{g (x)} g^{'} (x)$
Derivative of the general exponential function	$\frac{d}{d x} (b^{g (x)}) = b^{g (x)} g^{'} (x) ln b$
Derivative of the general logarithmic function	$\frac{d}{d x} ({log}_{b} g (x)) = \frac{g^{'} (x)}{g (x) ln b}$