Derivatives: Using binomial coefficients (see exercise 1) we can write for \(n\in\mathbb{N}\)
| \[ f(x) = x^n;\qquad f(x+h)=(x+h)^n=\left( \begin{array}{c} n\\0 \end{array} \right) x^n + \left( \begin{array}{c} n\\1 \end{array} \right)hx^{n-1} + \left( \begin{array}{c} n\\2 \end{array} \right) h^2x^{n-2}+... \] |
leading to
| \[ \frac{f(x+h)-f(x)}{h} = n\;x^{n-1}+\left( \begin{array}{c} n\\2 \end{array} \right)hx^{n-2}+...\,\rightarrow\,nx^{n-1}\quad\mbox{if $\quad h \rightarrow 0$} \] |
i.e.
| \[f(x)=x^n\rightarrow\,nx^{n-1}=f'(x)\quad n\in\mathbb{N}\] |
This already allows to calculate the derivatives of all functions which are defined by Taylor
series.
Plus:
| \[\begin{array}{ccl}\left[f(x)\pm g(x)\right]'&=&f'(x)\pm g'(x)\\\left[kf(x)\right]'&=&kf'(x)\end{array}\Rightarrow\;\mbox{''functions form vector-space''}\] |
Plus:
| \[\begin{array}{lccl} \mbox{product rule:}&\left[f(x)g(x)\right]'&=&f'(x)g(x)+f(x)g'(x)\\\\ \mbox{quotient rule:}&\left[\frac{f(x)}{g(x)}\right]'&=&\frac{f'(x)g(x)-f(x)g'(x)}{\left[g(x)\right]^2}\\\\ \mbox{chain rule:}&\left[f\left(g(x)\right)\right]'&=&g'(x)f'\left(g(x)\right) \end{array} \] |
To calculate the derivatives of inverse functions \(f^{-1}\) which are defined by
| \[x=f(f^{-1}(x))\] |
we can apply the chain rule:
| \[1 = \frac{d}{dx} f(f^{-1}(x))=\frac{df}{dx}\left(f^{-1}(x)\right)*\frac{df^{-1}}{dx}(x).\] |
For the natural logarithm which is the inverse of the exponential function, i.e.
| \[x=e^{\ln(x)}\] |
we find
| \[1 =\frac{de^x}{dx}\left(\ln(x)\right)*\frac{d\ln(x)}{dx}(x)=x\;\frac{d\ln(x)}{dx},\] |
i.e.
| \[\frac{d\ln(x)}{dx} =\frac{1}{x}.\] |
This now allows to calculate the derivative of \(f(x)=x^r\) with \(r \in \mathbb{R}\)
| \[\frac{dx^r}{dx}=\frac{de^{r\ln(x)}}{dx}=\frac{de^x}{dx}\left(r\ln(x)\right)*\frac{d\left(r \ln(x)\right)}{dx}=x^r \frac{r}{x}=rx^{r-1}\] |
in summary:
| \[ \left. \begin{array}{cclcccl} f(x)&=&x^r&\rightarrow&f'(x)&=&rx^{r-1},\;\;r\in\mathbb{R}\\ f(x)&=&e^x&\rightarrow&f'(x)&=&e^x\\ f(x)&=&\ln|x|&\rightarrow&f'(x)&=&\frac{1}{x}\quad!!\\ f(x)&=&\sin x&\rightarrow&f'(x)&=&\cos x\\ f(x)&=&\cos x&\rightarrow&f'(x)&=&-\sin x \end{array} \right\}\Rightarrow\,\mbox{all elementary functions!} \] |
Examples:
\begin{eqnarray*}f(x)=\sin(x\cdot\ln x)\,\Rightarrow\,f'(x)&=&(x\ln x)'\cos(x\ln x)\\&=&(1\ln x+x\frac{1}{x})\cos(x\ln x)=(\ln x+1)\cos(x\ln x)\end{eqnarray*}
| \[f(x)=e^{(e^x)}\Rightarrow\,(e^x)'e^{(e^x)}=e^xe^{(e^x)}=f'(x)\] |
| \[\begin{array}{cclcccl}f(x)&=&x^x&\Rightarrow&f'(x)&=&?\quad\mbox{no rule applicable}\\ &=&e^{x\ln x}&\Rightarrow&f'(x)&=&\left(\ln x+1\right)x^x\end{array}\] |
\(\Rightarrow\;\) Discussion of the functions we will have in the exercises!!
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© J. Carstensen (Math for MS)