Introduction to Differentiation
The gradient of a curve
In earlier work you found the gradient of a straight line using $\frac{\text{rise}}{\text{run}}$. A curve is harder because its steepness changes from point to point. Calculus solves this by measuring the gradient at a single point — the gradient of the tangent line that just touches the curve there. This instantaneous gradient is called the derivative.
Imagine a drone rising vertically, with its height modeled by $h(t) = 5t^2$ meters after $t$ seconds. Between $t = 2$ and $t = 3$ its average speed is $\frac{h(3)-h(2)}{3-2} = \frac{45-20}{1} = 25$ m/s. But how fast is it moving exactly at $t = 2$? If we shrink the time interval toward zero, the average speed settles onto a single value — the instantaneous speed. That limiting process defines the derivative.