Applications of Differentiation

Starting point: Years 9 and 10

Formulas you need

Speed, distance and time

$$\text{speed}=\frac{\text{distance}}{\text{time}}$$

On a distance-time graph, the gradient is the speed. On a velocity-time graph, the gradient is the acceleration.

Best values with quadratics

A quadratic with $a<0$ has a maximum at its vertex, $x=-\frac{b}{2a}$.

Quick summary

• A derivative is a rate of change. Its units are "$y$ per $x$".
• $v=\frac{dx}{dt}$ and $a=\frac{dv}{dt}$. At rest means $v=0$.
• For optimisation, write one formula in one variable, set the derivative to zero, and check the nature and the endpoints.
• A quadratic's best value is at its vertex, $x=-\frac{b}{2a}$.

Quiz