Math for Non-Geeks/ Examples for derivatives

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In this chapter we want to summarise the most important examples of derivatives. The derivative rules will allow us for computing derivatives of composite functions.

Table of important derivatives

In the following table $n\in \mathbb{N}$, $q\in \mathbb{Z}$ and $\tilde{n}\in {\mathbb{N}}_0$ is given. We also define $a,b,c\in ℝ$, $a_k\in ℝ$ and $p\in {ℝ}^{+}$.

function term	term of the derivative function	domain of definition of the derivative
$c$	$0$	$ℝ$
$x^n$	$n{x}^{n-1}$	$ℝ$
$ax+b$	$a$	$ℝ$
$a{x}^2+bx+c$	$2ax+b$	$ℝ$
$\sum _{k=0}^{\tilde{n}}{a}_k{x}^k$	$\sum _{k=1}^{\tilde{n}}{a}_{k}k{x}^{k-1}$	$ℝ$
$\frac{1}{x}$	$-\frac{1}{x^2}$	$ℝ\setminus \{0\}$
$x^{-n}=\frac{1}{x^n}$	$-\frac{n}{x^{n+1}}$	$ℝ\setminus \{0\}$
$x^q$	$q{x}^{q-1}$	$\{\begin{array}{cc}ℝ& q\in {\mathbb{Z}}_0^{+}\\ ℝ\setminus \{0\}& q\in {\mathbb{Z}}^{-}\end{array}$
$\sqrt{x}$	$\frac{1}{2\sqrt{x}}$	${ℝ}^{+}$
$\sqrt[n]{x}$	$\frac{1}{n\sqrt[n]{x^{n-1}}}$	${ℝ}^{+}$
$\sqrt[n]{x^q}$	$\frac{q}{n}\sqrt[n]{x^{q-n}}$	${ℝ}^{+}$
$\exp (x)=e^x$	$\exp (x)$	$ℝ$
$p^x=\exp (x\ln p)$	$\ln (p)\cdot p^x$	$ℝ$
$x^a=\exp (a\ln x)$	$a{x}^{a-1}$	$\{\begin{array}{cc}ℝ& a\ge 0\\ ℝ\setminus \{0\}& a<0\end{array}$
$\ln |x|$	$\frac{1}{x}$	$ℝ\setminus \{0\}$
${\log }_p|x|=\frac{\ln |x|}{\ln p}$	$\frac{1}{\ln (p)\cdot x}$	$ℝ\setminus \{0\}$
$\sin (x)$	$\cos (x)$	$ℝ$
$\cos (x)$	$-\sin (x)$	$ℝ$
$\tan (x)=\frac{\sin x}{\cos x}$	$\frac{1}{{\cos }^2(x)}=1+{\tan }^2(x)$	$ℝ\setminus \{\frac{\pi }{2}+k\pi |k\in \mathbb{Z}\}$
$\sec (x)=\frac{1}{\cos (x)}$	$\frac{\sin (x)}{{\cos }^2(x)}$	$ℝ\setminus \{\frac{\pi }{2}+k\pi |k\in \mathbb{Z}\}$
$\csc (x)=\frac{1}{\sin (x)}$	$-\frac{\cos (x)}{{\sin }^2(x)}$	$ℝ\setminus \{k\pi |k\in \mathbb{Z}\}$
$\cot (x)=\frac{\cos x}{\sin x}$	$-\frac{1}{{\sin }^2(x)}=-1-{\cot }^2(x)$	$ℝ\setminus \{k\pi |k\in \mathbb{Z}\}$
$\arcsin (x)$	$\frac{1}{\sqrt{1-x^2}}$	$(-1,1)$
$\arccos (x)$	$-\frac{1}{\sqrt{1-x^2}}$	$(-1,1)$
$\arctan (x)$	$\frac{1}{1+x^2}$	$ℝ$
$\text{arcot}(x)$	$-\frac{1}{1+x^2}$	$ℝ$
$\sinh (x)=\frac{e^x-e^{-x}}{2}$	$\cosh (x)$	$ℝ$
$\cosh (x)=\frac{e^x+e^{-x}}{2}$	$\sinh (x)$	$ℝ$
$\tanh (x)=\frac{\sinh x}{\cosh x}$	$\frac{1}{{\cosh }^2(x)}=1-{\tanh }^2(x)$	$ℝ$
$\mathrm{arsinh}(x)$	$\frac{1}{\sqrt{x^2+1}}$	$ℝ$
$\mathrm{arcosh}(x)$	$\frac{1}{\sqrt{x^2-1}}$	$(1,\mathrm{\infty })$
$\mathrm{artanh}(x)$	$\frac{1}{1-x^2}$	$(-1,1)$

Examples for computing derivatives

Now we will calculate some examples of derivatives from the table above. Often it comes down to determining the differential quotient of the function, i.e. a limit value. But sometimes it is also useful to use the calculation rules from the chapter before.

Constant functions

We start with some simple derivatives:

Power functions with natural numbers as powers

Now we turn to the derivative of power functions with natural powers. First we will deal with a few special cases:

Now we turn to the general case, i.e. the derivative of $x\mapsto x^n$ for $n\in \mathbb{N}$:

Polynomials and rational functions

Using the calculation rules for derivatives we can now calculate the derivatives of polynomial functions and rational functions:

Power functions with integer powers

We can already differentiate power functions with natural powers. Now we investigate those with negative integer exponents.

In the general case $x^{-n}=\frac{1}{x^n}$ with $n\in \mathbb{N}$ there is

Let us look again at the derivatives rule in the last case, i.e. $f^{\prime }(\tilde{x})=-\frac{n}{{\tilde{x}}^{n-1}}=-n{\tilde{x}}^{-n-1}$ for $n\in \mathbb{N}$. If we put $k=-n\in {\mathbb{Z}}^{-}$, we get $f^{\prime }(\tilde{x})=k{\tilde{x}}^{k-1}$. The derivative rule is hence the same as for $x^n$ with $n\in \mathbb{N}$. So we can summarize the two cases and get

Root functions

Now we investigate the derivative of root functions. We start again with the simplest case:

Now let us consider the general case of the $k$-th root function. Here there is

This can now be generalised

The (generalized) exponential function and generalized power functions

In this section we prove that the derivative of the exponential function is again the exponential function. So we can determine the derivative of the generalized exponential and power function.

Using the chain rule, the derivatives of the generalized exponential function $x\mapsto a^x$ for $a\in {ℝ}^{+}$ and the generalized power function $x\mapsto x^r$ for $r\in ℝ$ can be calculated:

Logarithmic functions

Now we turn to the derivative of the natural and generalised logarithm function. Since the natural logarithm is the inverse of the exponential function, we can deduce its derivative directly from rule for derivatives of inverse function:

The derivative can also be calculated directly using the differential quotient. If you want to try this, we recommend the corresponding exercise (missing).

Using the derivative of the natural logarithm function we can now immediately conclude

Trigonometric functions

Sine

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Cosine

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Tangent

The derivatives of secant and cosecant can be found in the corresponding exercise.

arc-functions

Using the rule for derivatives of the inverse function we can differentiate the arc-functions (which are inverses of sine, cosine, etc.)

arcsin and arccos

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arctan and arccot

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Hyperbolic functions

And finally, we determine the derivatives of the hyperbolic functions $\sinh$, $\cosh$ and $\tanh$:

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