What is a derivative? - Derivatives from scratch

The intuition: rate of change

Imagine a car that has covered a distance $s(t)$ by time $t$ . Its average speed between time $t$ and time $t + h$ is the distance traveled divided by the time elapsed:

v_{\text{avg}} = \frac{s(t + h) - s(t)}{h}

The shorter the interval $h$ , the closer you get to the speed at a single instant - the one the speedometer shows. The derivative is exactly this idea, applied to any function.

The definition

The derivative of a function $f$ at a point $x_0$ is the limit

f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h},

provided this limit exists and is finite. The expression inside the limit is called the difference quotient.

What it means on the graph

The difference quotient is the slope of the secant line through the points $(x_0, f(x_0))$ and $(x_0 + h, f(x_0 + h))$ . As $h$ approaches zero, the secant line turns into the tangent line. That’s why:

$f'(x_0)$ is the slope of the tangent line to the graph of $f$ at $x_0$ .

This gives you the equation of the tangent line:

y = f'(x_0)(x - x_0) + f(x_0)

Example: the derivative of $f(x) = x^2$

Compute the difference quotient at an arbitrary point $x$ :

\frac{(x + h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h

As $h \to 0$ , you’re left with $2x$ . So $f'(x) = 2x$ - for example, at $x = 3$ the tangent line to the parabola has slope $6$ .

The intuition: rate of change

The definition

What it means on the graph

Example: the derivative of f(x)=x2f(x) = x^2f(x)=x2

Sources

Example: the derivative of $f(x) = x^2$