The Derivative as a Slope

The derivative is ordinarily introduced as the slope of a curve. We chose not to do that because the polynomial approximations for functions are so important, and they give us a way of becoming used to power series long before we have to learn the formalities of those. However, it is truly more intuitive to think of the derivative as the slope of a function at a point. We can see it even in the definition: \[ f'(x_0)=\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}. \] You can see the change in the function divided by the distance over which the change occurred. If you think of \(f\) as a function describing the altitude of a road at any point, then you can see that the derivative is the rise of the road over the run. If the derivative were 0.1, that would correspond to a 10% grade. If the derivative were 1, it would be a 100% grade - a 45 degree road. If the derivative were -0.1, then the road would slope downhill at 10%.

The single trick here is in the limit. We are measuring the change in elevation of the function and dividing by the distance, but we are letting that distance become vanishingly small. The result is not any kind of average slope, but rather an instantaneous slope - the slope of the function at a single point.

Point x0 where tangent is shown:

Figure 1: A function, its derivative, and a line with the slope of the function.

In Figure 1 we can see a function, shown with a heavy blue curve; the graph of the derivative of that, shown with a light blue curve; and a line that has the same slope as the function itself. Whenever the derivative is positive, then the slope of the red line goes up. Whenever the derivative is negative the the line falls from left to right. Note that when e.g. \(x_0=-2.4\) the derivative curve crosses the \(x\)-axis, i.e. the derivative is zero. When the derivative is zero, the slope of the red line is zero - it is horizontal, flat. Does that happen anywhere else?

This "rise over run" property gave one of the creators of the differential calculus the idea of using a different notation for the derivative. Leibniz wrote the derivative of \(f\) at \(x_0\) using the notation \[\frac{df}{dx}(x_0).\] In other words, \[f'(x_0)=\frac{df}{dx}(x_0).\] The \(f'\) notation is due to Newton. You can see that \(f'\) emphasizes the new function that is defined as the derivative of a function \(f\), while \frac{df}{dx} emphasizes the derivative as a rate of change - it highlights that the derivative derives from the change in the function divided by the change in the argument.

Thus, we see that the derivative is important independent of its role in writing a polynomial approximation to a function: it gives an instantaneous rate of change for that function. If \(f(t)\) represents the position of a car at a time \(t\), then \(\frac{df}{dt}\) represents the car's velocity - how far it moves in a given time. If \(h(t)\) represents the height of a ball dropped from a building, then \(\frac{dh}{dt}\) represents how fast it is falling. If \(p(t)\) is the number of bacteria in a petri dish at time \(t\), then \(p'(t)\) tells how quickly that population is growing or shrinking.

Figure 2 shows the graph of a function in blue, and a red line through the point \(x_0=2\) whose slope is given by \[\frac{f(2+h)-f(2)}{2+h-2}=\frac{f(2+h)-f(2)}{h}.\] Thus, as \(h\to 0\) we see that the slope of the line approaches \(f'(2)\). Observe that as \(h\to 0\) the line seems to become tangent to the curve. When \(h\) actually equals zero, of course the quotient is undefined, so we lose our tangent line. That is why the derivative is defined in terms of a limit.

Value of h:

Figure 2: A function and a secant line whose slope approaches the derivative.