The derivative as a function

The slope of y = x² is −6 at x = −3, 0 at the bottom, and 6 at x = 3. One formula, f′(x) = 2x, gives all of them.

A slope for every x

In unit 1 you found f′(1) = 2 and f′(3) = 6 for f(x) = x², each by its own limit. It is more efficient to keep x as a letter. The derivative function is f′(x) = lim_h→0 (f(x + h) − f(x))/h, defined at every x where the limit exists. For x², ((x + h)² − x²)/h = (2xh + h²)/h = 2x + h, which approaches 2x. Now any slope is one substitution away: f′(−3) = −6, f′(0) = 0, f′(10) = 20. The derivative turns a function into a new function that reports the slope of the old one.

You can sketch f′ from the graph of f without any formula. Where f rises, its tangent slopes are positive, so f′ is above the x-axis. Where f falls, f′ is below. Where f has a smooth peak or valley, the tangent is horizontal and f′ crosses zero. For f(x) = x³ − 3x, the graph climbs until x = −1, falls until x = 1, then climbs again. So f′ is positive, then negative, then positive, with zeros at −1 and 1. The formula agrees: f′(x) = 3x² − 3, an upward parabola with roots ±1.

Where the derivative does not exist

A function can be continuous and still fail to have a derivative. The standard example is f(x) = |x| at 0. From the right the difference quotient is |h|/h = 1; from the left it is −1. The one-sided limits disagree, so f′(0) does not exist. The graph shows why: it has a sharp corner, and no single line is tangent there. OpenStax sums up the ways differentiability can fail at a point: a corner, a vertical tangent (as for the cube root of x at 0, where the slope grows without bound), more complicated behaviour such as that of x sin(1/x) at 0, and a discontinuity.

The last case connects to the limits course. If f is differentiable at a, then f is continuous at a. The reverse is false, and |x| is the proof: continuous at 0 but not differentiable. So differentiability is the stronger condition. A smooth graph with no corners, no vertical tangents and no breaks is differentiable everywhere on its domain; polynomials, sine, cosine and the exponential function all behave this way.

Derivatives of derivatives

Since f′ is a function, it has its own derivative, the second derivative f″. For x³ − 3x, f′(x) = 3x² − 3 and f″(x) = 6x. The second derivative measures how the slope itself is changing. If position is f, then f′ is velocity and f″ is acceleration, which unit 5 uses. You can keep going to f‴ and beyond, though the first two are the ones that appear most often.