Math for Non-Geeks/ Computing derivatives

{{#invoke:Math for Non-Geeks/Seite|oben}} In the last chapter we defined the derivative function $f^{\prime }:D\to ℝ$ of another differentiable function $f:D\to ℝ$ as follows: $f^{\prime }(\tilde{x}):=\underset{x\to \tilde{x}}{\lim }\frac{f(x)-f(\tilde{x})}{x-\tilde{x}}$. However, evaluating this limit can be a very cumbersome way to determine the derivative. For example, take the function $g:ℝ\to ℝ$ with $g(x)=x^2\cdot \ln (x)$. To calculate their derivatives we would have to determine $\underset{x\to \tilde{x}}{\lim }\frac{x^2\cdot \ln (x)-{\tilde{x}}^2\cdot \ln (\tilde{x})}{x-\tilde{x}}$ for every $\tilde{x}\in ℝ$.

It would be great to apply some rules to directly find an expression for the derivative function, which saves us the differential quotient computation. And luckily there are indeed derivative rules that trace derivatives of a complicated function back to derivatives of some very basic functions that are known exactly.

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If $f$ and $g$ are differentiable functions, with the compositions $af$ (with $a\in ℝ$), $f+g$, $fg$, $\frac{f}{g}$ and $f\circ g$ being all well-defined and differentiable, Then the following derivative rules apply:

Name	Regel
Factor rule	$(af{)}^{\prime }=a{f}^{\prime }$
Sum rule	$(f\pm g{)}^{\prime }=f^{\prime }\pm g^{\prime }$
Product rule	$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$
Quotient rule	$\left(\frac{f}{g}\right)=\frac{g{f}^{\prime }-f{g}^{\prime }}{g^2}$
Inverse rule	$\left(\frac{1}{g}\right)=-\frac{g^{\prime }}{g^2}$
Chain rule	$(f\circ g{)}^{\prime }=(f^{\prime }\circ g)\cdot g^{\prime }$
Special cases of the chain rule	$\begin{array}{cc}(f^n{)}^{\prime }& =n{f}^{n-1}\cdot f^{\prime }\\ (\sqrt{f}{)}^{\prime }& =\frac{f^{\prime }}{2\sqrt{f}}\\ (\exp \circ f{)}^{\prime }& =(\exp \circ f)\cdot f^{\prime }\\ (\ln \circ f{)}^{\prime }& =\frac{f^{\prime }}{f}\end{array}$
Inverse rule (yet missing)	$(f^{-1}{)}^{\prime }=\frac{1}{f^{\prime }\circ f^{-1}}$

All rules at one glance

The derivatives rules can be explained in simple words:

• Factor rule $(af{)}^{\prime }=a{f}^{\prime }$: The derivative is linear, so we can pull out any real (or even complex) number.
• Sum and difference rule $(f\pm g{)}^{\prime }=f^{\prime }\pm g^{\prime }$: The derivative is linear, so for a sum, we can take the derivative of both summands separately.
• Product rule $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$: "Derive the first function and the second remains unchanged plus derive the second function and the first remains unchanged".
• Quotient rule $\left(\frac{f}{g}\right)=\frac{g{f}^{\prime }-f{g}^{\prime }}{g^2}$: DDE-EDD is a simple memorization rule for the numerator ("denominator derivative enumerator minus enumerator derivative denominator")
• Inverse rule $\left(\frac{1}{g}\right)=-\frac{g^{\prime }}{g^2}$: This is the special case of the quotient rule with $f\equiv 1$ (enumerator is constant $1$).
• Chain rule $(f\circ g{)}^{\prime }=(f^{\prime }\circ g)\cdot g^{\prime }$: "Derive the outer function times derive the inner function". Caution, the derivative of the outer function must be taken with the inner function inserted ($f^{\prime }(g(x))$). The differentiation of the inner function must not be forgotten either.

Sum rule

Theorem

Now we want to determine the derivative of a function $g+f$, where $g:D\to ℝ$ and $f:D\to ℝ$ are both differentiable functions.

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{{#invoke:Math for Non-Geeks/Seite|unten|quellen=

• von Harten, G. Die Regeln der Differentialrechnung and ihre direkte Herleitung. Mitteilungen der Gesellschaft für Didaktik der Mathematik. Juli 2016, 8. PDF-Version.

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