Constant, power and factor rule
Constant rule
The constant rule states that the derivative of a constant is zero.
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Remember
A constant function $f(x)=c$ with $c\in\mathbb{R}$ has the derivative $f'(x)=0$
Examples
- $f(x)=8\rightarrow f'(x)=0$
- $f(x)=0,215\rightarrow f'(x)=0$
Power rule
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Remember
The derivative of a function $f(x)=x^\color{red}{n}$ is $f'(x)=\color{red}{n}\cdot x^{\color{red}{n}-1}$
Examples
- $f(x)=x^\color{red}{1}$
$f'(x)=\color{red}{1}\cdot x^{\color{red}{1}-1}=x^0=1$ - $f(x)=x^\color{red}{3}$
$f'(x)=\color{red}{3}\cdot x^{\color{red}{3}-1}=3x^2$ - $f(x)=x^\color{red}{-4}$
$f'(x)=\color{red}{-4}\cdot x^{\color{red}{-4}-1}=-4x^{-5}$
Factor rule
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Remember
The derivative of a function $f(x)=\color{red}{k}\cdot g(x)$ is $f'(x)=\color{red}{k}\cdot g'(x)$
Examples
Here, the power rule and the factor rule are applied:
- $f(x)=\color{red}{6}\cdot x^\color{blue}{7}$
$f'(x)=\color{red}{6}\cdot (\color{blue}{7}x^{\color{blue}{7}-1})=42x^6$ - $f(x)=\color{red}{3}\cdot x^\color{blue}{-1}$
$f'(x)=\color{red}{3}\cdot (\color{blue}{-1}x^{\color{blue}{-1}-1})=-3x^{-2}$