Derivatives of essential functions
The most important derivates of essential functions are listed here, to save you from having to derive them using theh-method.
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Hint
It is best to memorize the derivatives of these functions. Also, calculating the derivatives of complicated essential functions you should be able to use the derivative rules, especially the chain rule.
| Function | Derivative | |
|---|---|---|
| Derivative of power and square root function | ||
| $f(x)=x^n$ | $f'(x)=n\cdot x^{n-1}$ | |
| $f(x)=\sqrt{x}$ | $f'(x)=\frac{1}{2\sqrt{x}}$ | |
| Derivative of trigonometric functions | ||
| $f(x)=\sin(x)$ | $f'(x)=\cos(x)$ | |
| $f(x)=\cos(x)$ | $f'(x)=-\sin(x)$ | |
| $f(x)=\tan(x)$ | $f'(x)=\frac{1}{\cos^2(x)}$ | |
| Derivative of exponential functions | ||
| $f(x)=a^x$ | $f'(x)=a^x\cdot\ln(a)$ | |
| $f(x)=e^x$ | $f'(x)=e^x$ | |
| Derivative of logarithmic functions | ||
| $f(x)=\log_a(x)$ | $f'(x)=\frac{1}{x\cdot\ln(a)}$ | |
| $f(x)=\ln(x)$ | $f'(x)=\frac{1}{x}$ | |
Example
Differentiating with the chain rule
$f(x)=\sin(x^3+2x^2+3)$
Decompose function into parts
$g(x)=\sin(x)$ und $h(x)=\color{red}{x^3+2x^2+3}$Differentiate parts
Derivative of Sine and applying the Power rule
$g'(x)=\color{blue}{\cos}(x)$ and $h'(x)=\color{green}{3x^2+4x}$Inserting (Chain rule)
$f'(x)=\color{blue}{g'}(\color{red}{h(x)})\cdot \color{green}{h'(x)}$
$f'(x)$ $=\color{blue}{\cos}(\color{red}{x^3+2x^2+3})\cdot \color{green}{(3x^2+4x)}$