What is a derivative?

Functions are very simple. We give the function an input value and we get back exactly one output value, as long as the input was in the function’s domain. We usually say $y=f(x)$ and plot all the points ($x\text{,}$ $y$) on a graph. What if we instead tried graphing all the points ($x\text{,}$ $m$) where $m$ is the slope of the tangent line at $x$ on the original graph? Doing this turns out to be very useful. This idea is at the core of differential calculus.

The derivative of a function $f$ is another function, denoted by $f^{\prime }\text{,}$ which is like $f$ except it gives us $m_{\text{tan}}$ instead of $y\text{.}$ More specifically, $f^{\prime }(x)$ must be equal to the instantaneous rate of change on $f$ at $x\text{.}$ Before, we would talk about the slope of the tangent line (or the instantaneous rate of change) at $x=7$; now, we simply evaluate $f^{\prime }(7)\text{.}$ For example, if $f(x)=3x\text{,}$ then $f^{\prime }(x)$ will always give us 3 no matter what $x$ is.

It might seem like all we’re doing is changing our notation. Doesn’t the prime mark just mean, “give me the slope, not the y-value”? Well, yes, but there’s a bit more to it than that. Before we were using the first principle of calculus as a tool to find instantaneous rates of change. Now, we are doing this thing called differentiation to a function and getting an entirely new function called the derivative. Yes, $f^{\prime }$ is intimately related to $f\text{,}$ but don’t forget that it’s just a function. It can stand on its own, and we can do ordinary function stuff with it, like finding the zeros. We can even differentiate the derivative, giving us the second derivative of the original.

The mathematical definition of the derivative should look very familiar:

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}\text{.}$

Since the derivative is a function, we can and should graph it. Consider the simplest linear and quadratic functions—look at the slope of the original function and convince yourself that the derivative is correct:

One real-world example of a derivative is velocity, the derivative of position with respect to time. What do we mean when we say that a car is moving 20 m/s north? Maybe we observed the car taking 5 seconds to travel 100 metres. In any case, the only way we can measure velocity is by dividing a change in position by a change in time (the smaller, the better). If the derivative seems like an alien concept, just think of how ordinary it is to talk about the rate of change in position.