Math Derivative rules Constant, power and factor rule

# Constant, power and factor rule

## Constant rule

The constant rule states that the derivative of a constant is zero.

!

### 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

!

### 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

!

### 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:

• $f(x)=\color{red}{6}\cdot x^\color{blue}{7}$
$f'(x)=\color{red}{6}\cdot (\color{blue}{7}x^{\color{blue}{7}-1})=42x^6$

• $f(x)=\color{red}{3}\cdot x^\color{blue}{-1}$
$f'(x)=\color{red}{3}\cdot (\color{blue}{-1}x^{\color{blue}{-1}-1})=-3x^{-2}$