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 $f\text{'}(x_0)=m$. (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:

Again, we leave the verification of these statements to the reader.

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 $(0;0)$ is parallel to the $y$-axis and thus, it is not a graph of a function.

For $n\in \mathbb{N}$, root functions are described by $f(x)=x^{\frac{1}{n}}$. Of course, the differentiation rule given here also holds for $n=1$ or $n=-1$.

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

${\displaystyle f\text{'}(x)=p·x^{p-1}\hspace{0.5em}.}$