#### Chapter 7 Differential Calculus

Section 7.2 Standard Derivatives

# 7.2.2 Derivatives of Power Functions

In the last section, the derivative was introduced as the limit of the difference quotient. Accordingly, for a linear affine function (see Module 6, Section 6.2.4) $f:ℝ\to ℝ$, $x\to f\left(x\right)=mx+b$, where $m$ and $b$ are given numbers, we obtain for the derivative at the point ${x}_{0}$ the value $f\text{'}\left({x}_{0}\right)=m$. (Readers are invited to verify that fact themselves.)
For monomials ${x}^{n}$ with $n\ge 1$, it is easiest to determine the derivative using the difference quotient. Without any detailed calculation or any proof we state the following rules:
##### Derivative of ${x}^{n}$ 7.2.1
Let a natural number $n$ and a real number $r$ be given.
The constant function $f:ℝ\to ℝ$ with $x\to f\left(x\right):=r=r·{x}^{0}$ has the derivative $f\text{'}:ℝ\to ℝ$ with $x\to f\text{'}\left(x\right)=0$.
The function $f:ℝ\to ℝ$ with $x\to f\left(x\right):=r·{x}^{n}$ has the derivative

$f\text{'}:ℝ\to ℝ \text{with} x\to f\text{'}\left(x\right)=r·n·{x}^{n-1} .$

This differentiation rule is true for all $n\in ℝ\setminus \left\{0\right\}$.

Again, we leave the verification of these statements to the reader.
##### Example 7.2.2
Let us consider the function $f:ℝ\to ℝ$ with $x\to f\left(x\right)=5{x}^{3}$. According to the notation above, this is a function with $r=5$ and $n=3$. Thus for the value of the derivative at the point $x$ , we have

$f\text{'}\left(x\right)=5·3{x}^{3-1}=15{x}^{2} .$

For root functions, an equivalent statement holds. However, it should be noted that root functions are only differentiable for $x>0$ since the tangent line to the graph of the function at the point $\left(0;0\right)$ is parallel to the $y$-axis and thus, it is not a graph of a function.
##### Derivative of ${x}^{\frac{1}{n}}$ 7.2.3
For $n\in ℤ$ with $n\ne 0$, the function $f:\left[0;\infty \left[\text{}\to ℝ$, $x\to f\left(x\right):={x}^{\frac{1}{n}}$ is differentiable for $x>0$, and we have

$f\text{'}:\text{}\right]0;\infty \left[\text{}\to ℝ , x\to f\text{'}\left(x\right)=\frac{1}{n}·{x}^{\frac{1}{n}-1} .$

For $n\in ℕ$, root functions are described by $f\left(x\right)={x}^{\frac{1}{n}}$. Of course, the differentiation rule given here also holds for $n=1$ or $n=-1$.
##### Example 7.2.4
The root function $f:\left[0;\infty \left[\text{}\to ℝ$ with $x\to f\left(x\right):=\sqrt{x}={x}^{\frac{1}{2}}$ is differentiable for $x>0$. The value of the derivative at an arbitrary point $x>0$ is given by

$f\text{'}\left(x\right)=\frac{1}{2}·{x}^{\frac{1}{2}-1}=\frac{1}{2}·{x}^{-\frac{1}{2}}=\frac{1}{2·\sqrt{x}} .$

The derivative at the point ${x}_{0}=0$ does not exist since the slope of the tangent line to the graph of $f$ would be infinite there.

The tangent line to the graph of the given root function at the point $\left(1;1\right)$ has the slope $\frac{1}{2\sqrt{1}}=\frac{1}{2}$.

For $x>0$, the statements above can be extended to exponents $p\in ℝ$ with $p\ne 0$: The value $f\text{'}\left(x\right)$ of the derivative of the function $f$ with the mapping rule $f\left(x\right)={x}^{p}$ is, for $x>0$,

$f\text{'}\left(x\right)=p·{x}^{p-1} .$