derivatives / differentiation

Math Happens in calculus. It means the “rate of change” of something.

Definition of the derivative function

Let $f$ be a differentiatable function on open interval $I$. The function

$$\begin{array}{r}{f}^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}\end{array}$$

is the derivative of $f$ .

Notation: Let $y=f(x)$. The following notations all represent the derivative of $f$.

$$\begin{array}{r}{f}^{\prime }(x)=y^{\prime }=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}f=\frac{d}{dx}y\end{array}$$

Power rule

$$\begin{array}{r}\text{if}f(x)=x^n\text{ then}\\ \\ f^{\prime }(x)=n{x}^{n-1}\end{array}$$

The rules

Constant multiple rule

$$\begin{array}{rl}\frac{d}{dx}(cf(x))& =c\cdot f^{\prime }(x)\end{array}$$

sum/difference rule

$$\begin{array}{rl}\frac{d}{dx}(f(x)\pm g(x))& =f^{\prime }(x)\pm g^{\prime }(x)\end{array}$$

constant rule

$$\begin{array}{rl}\frac{d}{dx}(c)& =0\end{array}$$

product rule

$$\begin{array}{rl}\frac{d}{dx}(f(x)g(x))& =f(x)g^{\prime }(x)+f^{\prime }(x)g(x)\end{array}$$

quotient rule

$$\begin{array}{rl}\frac{d}{dx}(\frac{f(x)}{g(x)})& =\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{g(x{)}^2}\end{array}$$

power rule

$$\begin{array}{rl}\frac{d}{dx}(x^n)& =n\cdot x^{n-1}\end{array}$$

generalized power rule

$$\begin{array}{rl}\frac{d}{dx}(g(x{)}^n)& =n\cdot (g(x){)}^{n-1}\cdot g^{\prime }(x)\end{array}$$

Chain rule

$$\begin{array}{rl}f(g(x))& =f^{\prime }(g(x))\cdot g^{\prime }(x)\end{array}$$

Derivative of trigonometric functions

$$\begin{array}{rl}\frac{d}{dx}(\sin x)& =\cos x\\ & \\ \frac{d}{dx}(\cos x)& =-\sin x\\ & \\ \frac{d}{dx}(\tan x)& ={\sec }^{2}x\\ & \\ \frac{d}{dx}(\csc x)& =\sec x\tan x\\ & \\ \frac{d}{dx}(\cot x)& =-{\csc }^{2}x\end{array}$$

derivative of exponential functions

where a > 0 & a != 1

$$\begin{array}{rl}\frac{d}{dx}(a^x)& =\ln a\cdot a^x\\ & \\ \frac{d}{dx}(e^x)& =e^x\end{array}$$

others I don’t have a name for

$$\begin{array}{rl}\frac{d}{dx}(\ln x)& =\frac{1}{x}\\ & \\ \frac{d}{dx}(x)& =1\end{array}$$