Key Equations

Key Equations

Difference quotient	$Q=\frac{f\left(x\right)-f(a)}{x-a}$
Difference quotient with increment $h$	$Q=\frac{f\left(a+h\right)-f(a)}{a+h-a}=\frac{f\left(a+h\right)-f(a)}{h}$
Slope of tangent line	$m_{\text{tan}}=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f(a)}{x-a}$ $m_{\text{tan}}=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f(a)}{h}$
Derivative of $f(x)$ at $a$	$f^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$ $f^{\prime }(a)=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f(a)}{h}$
Average velocity	$v_{a\text{ve}}=\frac{s\left(t\right)-s(a)}{t-a}$
Instantaneous velocity	$v\left(a\right)=s^{\prime }\left(a\right)=\underset{t\to a}{\text{lim}}\frac{s\left(t\right)-s(a)}{t-a}$

The derivative function

$f^{\prime }\left(x\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

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

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

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

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