The derivative as a function

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Here we study the derivative of a function, as a function, in its own right.

The derivative of a function, as a function

We know that to find the derivative of a function at a point $x=a$ we write $f'(a) = \lim _{h\to 0}\frac {f(a+h)-f(a)}{h}.$ However, if we replace the given number $a$ with a variable $x$ , we now have $f'(x) = \lim _{h\to 0}\frac {f(x+h)-f(x)}{h}.$ This tells us the instantaneous rate of change at any given point $x$ .

The notation:

$f'(a)$ means take the derivative of $f$ first, then evaluate at $x=a$ .

In other words, given $f$ a function of $x$ $f'(a) = \eval {\ddx f(x)}_{x=a}.$

Given a function $f$ from the real numbers to the real numbers, the derivative $f'$ is also a function from the real numbers to the real numbers. Understanding the relationship between the functions $f$ and $f'$ helps us understand any situation (real or imagined) involving changing values.

Let $f(x) = 3x+2$ . What is $f'(-1)$ ?

$f'(-1) = 0$ because $f'(3)$ is a number, and a number corresponds to a horizontal line, which has a slope of zero. $f'(-1) = 3$ because $y=f(x)$ is a line with slope $3$ . We cannot solve this problem yet.

Here we see the graph of $f'$ .

Describe $y=f(x)$ when $f'$ is positive. Describe $y=f(x)$ when $f'$ is negative. When $f'$ is positive, $y=f(x)$ is positiveincreasingnegativedecreasing . When $f'$ is negative, $y=f(x)$ is positiveincreasingnegativedecreasing

Which of the following graphs could be $y = f(x)$ ?

The derivative as a function of functions

While writing $f'$ is viewing the derivative of $f$ as a function in its own right, the derivative itself $\ddx$ is in fact a function that maps functions to functions,

$\begin{align*} \ddx x^2 &= 2x\\ \ddx f(x) &= f'(x). \end{align*}$

As a function, is $\ddx$ one-to-one?

yes no

Many different functions share the same derivative since the derivative records only the slope of the tangent line and not the value, or height of the function.