What you need to know
Differentiation measures rate of change. In graph questions it gives the gradient of the curve, and setting the derivative to zero locates stationary points that may be maxima, minima or other turning behaviour.
- Differentiate powers term by term.
- Evaluate the derivative at a given x-value to find a gradient.
- Set the derivative equal to zero to locate stationary points.
- Use a second derivative or sign change where required to classify a stationary point.
Key mathematics
Power rule
\frac{d}{dx}(x^n)=nx^{n-1}
Apply to each power term.Stationary point condition
\frac{dy}{dx}=0
Solve for x, then substitute into the original function for y.Second derivative test
\frac{d^2y}{dx^2}>0\Rightarrow\text{local minimum}
A negative second derivative indicates a local maximum.Worked example
For $y=x^3-3x^2+2$, $$\frac{dy}{dx}=3x^2-6x=3x(x-2).$$ Stationary points occur at $x=0$ and $x=2$. Substitute into the original function to find their coordinates before classifying them.
EXAM TECHNIQUE
Exam tips
- Differentiate first, then substitute the x-value when finding a gradient.
- For stationary points, calculate both x and y coordinates.
- Keep differentiation notation clear so the examiner can follow your method.
WATCH OUT
Common mistakes
- Substituting into the original function before differentiating when a gradient is required.
- Solving dy/dx=0 but forgetting to find the corresponding y-values.
- Reducing the power without multiplying by the original exponent.
Frequently asked questions
What does the derivative represent?
It represents the instantaneous rate of change and, geometrically, the gradient of the tangent to a curve.
How do I find stationary points?
Set the first derivative equal to zero, solve for x and substitute back into the original function.
How can I classify a stationary point?
Use the second derivative or examine how the first derivative changes sign around the point.