2.2: The 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 $a$ be any number in the domain of $f(x)$, 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.

Functions you know about look like $f(x) = x + 1$, where it has a input-output structure.

Both input and output are real numbers.

Because the derivative is a function, the input is a function and the output is also a function!

Given this graph

Sketch the derivative.

If $f(x) = x^3 - x$, 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.

Where is the function $f(x) = \lvert x \rvert$ differentiable?

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 reason why this is true (often called a mathematical proof) is in the textbook (page 122); we will omit it here.

Higher-Order Derivatives

If $f(x)$ is a function, $f'(x)$ is also a function.

$f'(x)$ might have a derivative of it's own!

To find the second derivative, first find $f'(x)$, then find the derivative of $f'(x)$.

In general, this extends to the $n$th derivative. Here are common notations for multiple derivatives:

$f'(x), f''(x), f'''(x), \cdots, f^{(n)}(x)$
$D_xf(x), D^2_xf(x), D^3_xf(x), \cdots, D^{(n)}_xf(x)$
$y', y'', y''', \cdots, y^{(n)}$
$\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}, \dfrac{d^3y}{dx^3}, \cdots, \dfrac{d^ny}{dx^n}$

The last notation is called Leibniz notation.

If $f(x) = x^3 - x$, find $f''(x)$ and interpret the meaning of $f''(x)$.

We can interpret the second derivative as the rate of change of a rate of change.

A familiar example of a second derivative is acceleration.

If you have a position function $s(t)$, then we saw $v(t) = s'(t)$ is the velocity.

The rate of change of velocity is acceleration, or in symbols: \[a(t) = v'(t) = s''(t)\]