A-level Maths · Maths
Differentiation: a clear study guide
Differentiation describes how a function changes. Learn the power rule, then connect the derivative to gradients, stationary points and the shape of a graph.
The power rule
For a term ax^n, differentiate by multiplying by n and reducing the power by one: d(ax^n)/dx = anx^(n-1). Constants differentiate to zero because they do not change as x changes.
Differentiate each term separately, keep brackets organised and simplify only after applying the rule.
- The derivative gives the gradient function.
- A constant has derivative zero.
- Check the power and coefficient after each term.
Stationary points
A stationary point occurs where the gradient is zero, so solve f'(x) = 0 and substitute the x-values into the original function to find coordinates. The second derivative can help classify a local maximum or minimum when the method is allowed.
A derivative can also be used to find where a function is increasing or decreasing by studying the sign of f'(x).
Interpret the result
Write a conclusion in the language of the question. A gradient may represent a rate of change, while a stationary point may represent a turning point or an optimum. Include coordinates and units where they are relevant.
Quick check
What is the derivative of 5x^3?
Answer: 15x^2.
What equation identifies stationary points?
Answer: Set the first derivative equal to zero: f'(x) = 0.
Common questions
What does a derivative represent?
It represents the gradient or instantaneous rate of change of a function, depending on the context.
How do I find the coordinates of a stationary point?
Solve f'(x) = 0 for x, then substitute each x-value into the original function to find y.