Derivatives

## First definition

Let $d d x : ℚ x → ℚ x$ such that if $\beta ,\gamma \in ℚ$ and $f,g\in ℚ[[x]]$ then

1. $\frac{d}{dx}\left(\beta f+\gamma g\right)=\beta \frac{df}{dx}+\gamma \frac{dg}{dx}$,
2. $\frac{d\left(fg\right)}{dx}=f\frac{dg}{dx}+\frac{df}{dx}g$, and
3. $\frac{d}{dx}\left(x\right)=1$.

## Second definition

Let $f:\left[a,b\right]\to ℝ$. The derivative of $f$ at $x=c$ is $f ′ c = lim x → c f x - f c x - c$ or equivalently $f ′ c = lim Δ x → 0 f c + Δ x - f c Δ x$

## Theorem

Let $f:\left[a,b\right]\to ℝ$ and $g:\left[a,b\right]\to ℝ$ and let $\beta ,\gamma \in ℝ$. Assume that $f\prime \left(c\right)$ and $g\prime \left(c\right)$. Then

1. $\left(\beta f+\gamma g\right)\prime \left(c\right)=\beta f\prime \left(c\right)+\gamma g\prime \left(c\right)$,
2. $\left(fg\right)\prime \left(c\right)=f\prime \left(c\right)g\left(c\right)+f\left(c\right)g\prime \left(c\right)$,
3. if $f:\left[a,b\right]\to ℝ$ is given by $f\left(x\right)=x$ then $f\prime \left(c\right)=1$, and
4. if $f\prime \left(c\right)$ exists then $f$ is continuous at $x=c$.

 Proof. (d): By assumption $f\prime \left(c\right)$ exists. So $\underset{x\to c}{\mathrm{lim}}\frac{f\left(x\right)-f\left(c\right)}{x-c}$ exists. There exists $l$ such that $\underset{x\to c}{\mathrm{lim}}\frac{f\left(x\right)-f\left(c\right)}{x-c}$. To show: $f$ is continuous at $x=c$. To Show: $\underset{x\to c}{\mathrm{lim}}f\left(c\right)=f\left(c\right)$. To show: If $ϵ\in {ℝ}_{>0}$ then there exists $\delta \in {ℝ}_{>0}$ such that if $\left|x-c\right|<\delta$ then $\left|f\left(x\right)-f\left(c\right)\right|<ϵ$. Assume $ϵ\in {ℝ}_{>0}$. We know that there exists ${\delta }_{1}\in {ℝ}_{>0}$ such that if $\left|x-c\right|<{\delta }_{1}$, then $\left|\frac{f\left(x\right)-f\left(c\right)}{x-c}-l\right|<\frac{ϵ}{2}$. Let $\delta =\mathrm{min}\left(1,{\delta }_{1},\frac{ϵ}{2l}\right)$. To show: If $\left|x-c\right|<\delta$ then $\left|f\left(x\right)-f\left(c\right)\right|$. Assume $\left|x-c\right|<\delta$. To show: $\left|f\left(x\right)-f\left(c\right)\right|<ϵ$. $\begin{array}{rcl}\left|f\left(x\right)-f\left(c\right)\right|& =& \left|\frac{f\left(x\right)-f\left(c\right)}{x-c}\left(x-c\right)\right|\\ & =& \left|\left(\frac{f\left(x\right)-f\left(c\right)}{x-c}-l\right)\left(x-c\right)+l\left(x-c\right)\right|\\ & \le & \left|\frac{f\left(x\right)-f\left(c\right)}{x-c}-l\right|\left|x-c\right|+l\left|x-c\right|\\ & <& \frac{ϵ}{2}·\delta +l\delta \\ & <& \frac{ϵ}{2}+\frac{lϵ}{2l}\\ & =& ϵ\text{.}\end{array}$ $\square$

## Standard derivatives

1. If $n\in ℤ$ then $\frac{d}{dx}{x}^{n}=n{x}^{n-1}$.
2. If $c\in ℂ$ then $\frac{d}{dx}{x}^{c}=c{x}^{c-1}$.
3. $\frac{d}{dx}{e}^{x}={e}^{x}$.
4. $\frac{d}{dx}\mathrm{log}x=\frac{1}{x}$.
5. $\frac{d}{dx}\mathrm{sin}x=cosx$.
6. $\frac{d}{dx}\mathrm{arcsin}x=\frac{-1}{\sqrt{1-{x}^{2}}}$.

## The chain rule

$\frac{d}{dx}\left(f\circ g\right)=\frac{df}{dg}\frac{dg}{dx}$.