Physics equations/01-Introduction/A:reviewCALCULUS

Calculus
If f and g are functions of x and a and b are constants, then:  $$\frac{d}{dx} x^n = nx^{n-1}.$$

$$ \frac{d(af+bg)}{dx} = a\frac{df}{dx} +b\frac{dg}{dx}.$$     $$\frac{d(fg)}{dx} = \frac{df}{dx} g + f \frac{dg}{dx}.$$

$$\frac{dh}{dx} = \frac{dh}{dg} \frac{dg}{dx}.$$    $$\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}.$$

If y=y(x) and x=x(y) are inverse functions then: $$ \frac{dx}{dy} = \frac{1}{dy/dx}.$$

Indefinite integrals, where $$C$$ is the arbitrary constant of integration:


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

Exponential and trigonometric functions
If a is a constant, then: $$ \frac{d}{dx}\left(e^{ax}\right) = ae^{ax}.$$    $$ \frac{d}{dx}\left( \ln x\right)  = {1 \over x} ,\quad x \ne 0$$     $$\Rightarrow(\ln f)'= \frac{f'}{f} \quad$$ wherever f is positive.
 * {| style="width:100%; background:transparent; margin-left:2em;"


 * width=50%|$$ (\sin ax)' = a\cos x \,$$
 * width=50%|$$ (\arcsin x)' = { 1 \over \sqrt{1 - x^2}} \,$$
 * $$ (\cos ax)' = -a\sin x \,$$
 * $$ (\arccos x)' = -{1 \over \sqrt{1 - x^2}} \,$$
 * $$ (\tan x)' = \sec^2 x = { 1 \over \cos^2 x} = 1 + \tan^2 x \,$$
 * $$ (\arctan x)' = { 1 \over 1 + x^2} \,$$
 * $$ (\sec x)' = \sec x \tan x \,$$
 * $$ (\operatorname{arcsec} x)' = { 1 \over |x|\sqrt{x^2 - 1}} \,$$
 * $$ (\csc x)' = -\csc x \cot x \,$$
 * $$ (\operatorname{arccsc} x)' = -{1 \over |x|\sqrt{x^2 - 1}} \,$$
 * $$ (\cot x)' = -\csc^2 x = { -1 \over \sin^2 x} = -(1 + \cot^2 x)\,$$
 * $$ (\operatorname{arccot} x)' = -{1 \over 1 + x^2} \,$$
 * }
 * $$ (\operatorname{arccsc} x)' = -{1 \over |x|\sqrt{x^2 - 1}} \,$$
 * $$ (\cot x)' = -\csc^2 x = { -1 \over \sin^2 x} = -(1 + \cot^2 x)\,$$
 * $$ (\operatorname{arccot} x)' = -{1 \over 1 + x^2} \,$$
 * }
 * }

Fundamental theorem of calculus

 * $$\int_{a}^b \frac {dF}{ds}\,ds= \int dF = F|_a^b=F(b)-F(a)$$


 * $$\Rightarrow \text{If }\;F(x)=\int_{a}^x f(s)\,ds,\,$$    $$\text{ then }\;\frac {dF}{dx}=f(x)$$

Taylor series and Euler's equations

 * $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots $$


 * $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$


 * $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots $$


 * $$ \Rightarrow \; e^{i \theta} = \cos \theta + i\sin\theta$$


 * Derivative
 * Differential calculus
 * Vector calculus identities
 * Differentiable function
 * Differential of a function
 * Limit of a function
 * Function
 * List of mathematical functions
 * Trigonometric functions
 * Inverse trigonometric functions
 * Hyperbolic functions
 * Inverse hyperbolic functions
 * Matrix calculus
 * Differentiation under the integral sign