3.2: Derivative as a Function

Recall the definition of a derivative at $x = a$:

The derivative of a function $f$ at a number $a$, denoted by $f'(a)$, is \[f'(a) = \lim_{h\rightarrow 0} \dfrac{f(a + h) - f(a)}{h}\] if this limit exists.
If we let $a$ vary, we can think of the derivative as a function:
The derivative of a function $f$, denoted by $f'(x)$, is \[f'(x) = \lim_{h\rightarrow 0} \dfrac{f(x + h) - f(x)}{h}\] if this limit exists.
Given this graph Sketch the derivative.
If $f(x) = \dfrac{1}{x}$, find $f'(x)$.
If $f(x) = \sqrt{x}$, find $f'(x)$. Compare their graphs.

Differentiability

If you let $y = f(x)$, the derivative has many different notations: \[f'(x) = y' = \dfrac{dy}{dx} = \dfrac{d}{dx} f(x) = Df(x) = D_xf(x)\]

A function $f$ is differentiable at $a$ if $f'(a)$ exists. It is differentiable on an open interval $(a,b)$ if it is differentiable at every number in the interval.
Show $f(x) = \lvert x \rvert$ is not differentiable at $x = 0$.

There are three cases when $f'(x)$ does not exist.

How are differentiability and continuity related?

If $f$ is differentiable at $a$, then $f$ is continuous at $a$.

The proof is in the textbook; we will omit it here.

The metabolic power $P(v)$ (amount of energy required) consumed by humans who run then walk at a speed $v$ is shown below
  1. Is $P$ a differentiable function of $v$?
  2. Sketch a graph of $P'(v)$.