Derivative

Derivative for single variable functions

$f$ is differentiable at a if $\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$ exists. $f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

• product rule: $(fg{)}^{\prime }=f{g}^{\prime }+f^{\prime }g$
• chain rule: $(f\circ g{)}^{\prime }(a)=f^{\prime }(g(a))g^{\prime }(a)$

Mean Value Theorem

Let $f:[a,b]\to \mathbb{R}$ be a continuous function on $[a,b]$, and differentiable function on $(a,b)$, then there exists at least one $c\in (a,b)$ such that

f’(c) = \frac{f(b) - f(a)}{b - a}.

> **Note:** Rolle's Theorem is a special case of the Mean Value Theorem when $f(a) = f(b).$ That is $f'(c) = 0$. > [!theorem] Cauchy Mean Value Theorem (CMVT) > > Let $f, g : [a,b] \to \mathbb{R}$ be continuous functions on $[a,b]$, and differentiable functions on $(a,b)$, then there exists $c \in (a,b)$ such that > $$ > (f(b)-f(a))g'(c)\ =\ (g(b)-g(a))f'(c). > $$ > **Note:** If \(g(x) = x\), the Cauchy MVT reduces to the standard Mean Value Theorem. >[!definition] > > - $f$ is increasing $a<b \implies f(a)<f(b)$. > - $f$ is decreasing $a<b \implies f(a)>f(b)$. > - $x$ is a local maximum point if $\exists \delta$ such that $f(x) \geq f(y)$ $\forall y \in (x-\delta, x+\delta)$. > - $x$ is a local minimum point if $\exists \delta$ such that $f(x) \leq f(y)$ $\forall y \in (x-\delta, x+\delta)$. > - If $f'(x) = 0$ we say that $x$ is a critical point of $f$. >[!proposition] > >- If $x$ is an extreme point for $f$ and $f$ is differentiable at $x$ then $f'(x) = 0$ >- At an extreme point, either the derivative vanishes or doesn't exist. >- Suppose $f'(a) = 0$ and $f''(a) >0$ then $f$ has a local min at a.