Applications of Differentiation

Basics and Definitions

Differentiation can be used to find rates of change. The derivative of a function gives the rate at which the function changes per unit increase in the input.
Tangents and normals to a curve: The derivative at a particular point gives the slope of the tangent to the curve at that point.
Stationary points on a curve occur where the derivative equals zero. There are three types of stationary points: maximum points, minimum points and points of inflection.

Optimisation Problems

Differentiation is used in optimisation problems where we need to maximise or minimise a quantity.
One common application is in business, for cost minimisation or profit maximisation.
To solve these problems, derive a formula for the quantity to be maximised or minimised, then differentiate the function and set it equal to zero.

Differentiation can be used to examine how quantities change over time. One common usage is in physics for analysing motion.
Velocity and acceleration, key concepts in kinematics, can be found by differentiating the displacement function once for velocity, and twice for acceleration.
Newton’s laws of motion rely on applications of differentiation.

Differentiation can be used to sketch curves.
Knowing the derivative of a function can reveal where the function is increasing and decreasing, the positions of maximum and minimum points, and the shape of the curve.
This information can be used in various fields such as engineering model systems or economists analysing trends.