Your first derivative

The derivative sounds like machinery, but it answers a plain question: how steep is this curve right here?

From average to instant

On a line, steepness is rise over run. On a curve the slope changes, so pick two points and measure the line between them:

$$\frac{f(x+h) - f(x)}{h}$$

For $f(x) = x^2$:

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

Now let $h$ shrink toward zero. The leftover $h$ vanishes, and the slope at $x$ is exactly $2x$.

A rule that saves time

Constants disappear and sums split, so differentiating term by term is allowed.

Two slips to avoid

See it happen

Squeeze two points together and watch the secant line turn into the tangent in the Derivatives as Slope explainer.