Defining the derivative

For f(x) = x², the slope of the secant from x = 1 to x = 1 + h works out to exactly 2 + h. Let h shrink to nothing and 2 is all that is left.

The difference quotient

The limits course found tangent slopes by taking secants through a fixed point P = (a, f(a)) and a moving point Q, and letting Q slide toward P. It is convenient to name the gap between the two x-values h, so that Q = (a + h, f(a + h)). The slope of the secant is then the difference quotient (f(a + h) − f(a))/h. It is an ordinary fraction, defined for every h except 0, and it measures the average rate of change of f between a and a + h.

OpenStax defines the derivative of f at a as the limit of this quotient: f′(a) = lim_h→0 (f(a + h) − f(a))/h, provided the limit exists. It is read f prime of a. Geometrically it is the slope of the tangent line at (a, f(a)); as a rate, it is the instantaneous rate of change of f at a. If the limit exists, f is differentiable at a. The tangent line itself is then y = f(a) + f′(a)(x − a), the point-slope form with the point (a, f(a)) and slope f′(a).

Working one out

Take f(x) = x² and a = 1. Then f(1 + h) = 1 + 2h + h², so f(1 + h) − f(1) = 2h + h², and dividing by h gives 2 + h. This simplification is the whole trick: the troublesome h in the denominator cancels, and what remains can be evaluated at h = 0. So f′(1) = 2. The tangent line at (1, 1) is y = 1 + 2(x − 1), or y = 2x − 1. With h = 0.1 the secant slope is 2.1, with h = 0.01 it is 2.01, which matches the table you built by hand in the limits course.

The same steps work for other functions. For f(x) = 1/x at a = 1, the difference quotient is (1/(1 + h) − 1)/h. Combine the top over a common denominator: (1 − (1 + h))/(1 + h) = −h/(1 + h). Divide by h to get −1/(1 + h), which approaches −1. So the slope of 1/x at x = 1 is −1, and the curve is falling there. For a line f(x) = mx + b, the quotient is (m(a + h) + b − ma − b)/h = m for every h, so f′(a) = m: the derivative of a line is its slope, as it should be.

There are several ways to write a derivative. Besides f′(a), Leibniz's notation dy/dx keeps the picture of a small change in y over a small change in x, and it is the form you will see most often in science. Units follow from that picture. If s is a distance in meters and t a time in seconds, ds/dt is in meters per second. A derivative always carries units of output per unit of input, which is a good check on any answer.