Calculus I

Basic Functions and Formulas

Logarithmic functions	$\displaystyle\log_{a} x = \frac{\ln x}{\ln a}$
Exponential functions	$\displaystyle a^x = e^{x\ln a}$
Trigonometric functions	$\displaystyle\sec x = \frac{1}{\cos x}$ ; $\displaystyle\csc x = \frac{1}{\sin x}$ $\displaystyle\tan x = \frac{\sin x}{\cos x}$ ; $\displaystyle\cot x = \frac{\cos x}{\sin x}$ $\displaystyle\cos^2 x + \sin^2 x = 1$ $\displaystyle 1 + \tan^2 x = \sec^2 x$ ; $\displaystyle 1 + \cot^2 x = \csc^2 x$ $\sin(u + v) = \sin u \cos v + \cos u \sin v$ $\cos(u + v) = \cos u \cos v - \sin u \sin v$ $\displaystyle\sin^2 x = \frac{1 - \cos 2x}{2}$ $\displaystyle\cos^2 x = \frac{1 + \cos 2x}{2}$
Hyperbolic functions	$\displaystyle\sinh x = \frac{e^x - e^{-x}}{2}$ $\displaystyle\cosh x = \frac{e^x + e^{-x}}{2}$ $\displaystyle\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ $\displaystyle\cosh^2 x - \sinh^2 x = 1$ $\displaystyle\sinh^{-1} x = \ln\left\vert x + \sqrt{x^2 + 1}\right\vert$ $\displaystyle\cosh^{-1} x = \ln\left\vert x + \sqrt{x^2 - 1}\right\vert$ $\displaystyle\tanh^{-1} x = \frac{1}{2}\ln\left\vert\frac{1 + x}{1 - x}\right\vert$