Derivatives of Elementary Functions

Theorem: Power Rule

$$(x^{\alpha }{)}^{\prime }=\alpha x^{\alpha -1}\forall \alpha \in \mathbb{R}$$

PROOF

TODO

Theorem: Derivative of Exponential Functions

$$({\mathrm{e}}^x{)}^{\prime }={\mathrm{e}}^x$$ $$(a^x{)}^{\prime }=a^x\ln a$$

PROOF

$$\begin{array}{rl}(e^x{)}^{\prime }& =\underset{\Delta x\to 0}{\lim }\frac{e^{x+\Delta x}-e^x}{\Delta x}=\underset{\Delta x\to 0}{\lim }\frac{e^x(e^{\Delta x}-1)}{\Delta x}=e^x\underset{\Delta x\to 0}{\lim }\frac{e^{\Delta x}-1}{\Delta x}\\ & =e^x\underset{\Delta x\to 0}{\lim }\frac{\underset{n\to \infty }{\lim }{\left(1+\frac{\Delta x}{n}\right)}^n-1}{\Delta x}=e^x\underset{\Delta x\to 0}{\lim }\underset{n\to \infty }{\lim }\frac{(1+\frac{\Delta x}{n}{)}^n-1}{\Delta x}\\ & =e^x\underset{\Delta x\to 0}{\lim }\underset{n\to \infty }{\lim }\frac{1+n\frac{\Delta x}{n}+\left(\begin{array}{c}n\\ 2\end{array}\right){\left(\frac{\Delta x}{n}\right)}^2+\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right){\left(\frac{\Delta x}{n}\right)}^{n-1}+{\left(\frac{\Delta x}{n}\right)}^n-1}{\Delta x}\\ & =e^x\underset{\Delta x\to 0}{\lim }\underset{n\to \infty }{\lim }\left(1+\left(\begin{array}{c}n\\ 2\end{array}\right)\frac{\Delta x}{n^2}+\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right)\frac{\Delta x^{n-2}}{n^{n-1}}+\frac{\Delta x^{n-1}}{n^n}\right)\\ & =e^x\underset{n\to \infty }{\lim }\underset{\Delta x\to 0}{\lim }\left(1+\underset{\to 0}{\underbrace{\left(\begin{array}{c}n\\ 2\end{array}\right)\frac{\Delta x}{n^2}+\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right)\frac{\Delta x^{n-2}}{n^{n-1}}+\frac{\Delta x^{n-1}}{n^n}}}\right)\\ & =e^x\underset{n\to \infty }{\lim }1=e^x\end{array}$$ $$\begin{array}{r}(a^x{)}^{\prime }=((e^{\ln a}{)}^x{)}^{\prime }=(e^{x\ln a}{)}^{\prime }=e^{x\ln a}\ln a=(e^{\ln a}{)}^x\ln a=a^x\ln a\end{array}$$

Theorem: Derivative of Logarithmic Functions

$$(\ln x{)}^{\prime }=\frac{1}{x}$$ $$({\log }_{a}x{)}^{\prime }=\frac{1}{x\ln a}$$

PROOF

TODO