TMUA Differentiation and Stationary Points

Rules

For any rational $n$, $\dfrac{d}{dx}x^n=nx^{n-1}$. Differentiate sums term by term, and simplify first, for example rewriting $\dfrac{1}{x}$ as $x^{-1}$ and $\sqrt x$ as $x^{1/2}$.

Tangents and normals

The tangent at a point has gradient equal to the derivative there. The normal is perpendicular to the tangent.

Stationary points

Solve $f'(x)=0$. If $f''(x)>0$ it is a minimum, if $f''(x)<0$ a maximum. If $f''(x)=0$, check the sign of $f'$ either side.

Increasing and decreasing functions

A function is increasing where $f'(x)>0$ and decreasing where $f'(x)<0$. Questions can combine this with algebra: a cubic has two stationary points exactly when $f'(x)=0$ has two distinct roots, so the discriminant of $f'$ must be positive.

The specification treats points of inflexion qualitatively only: they are where the curve changes from bending one way to the other.

Always check what the question wants: the coordinates, the value of $y$, the nature of the point, or an equation of a line.

Worked examples

Common mistakes

• Giving the $x$-value of a stationary point when the question asks for the value of $y$.
• Differentiating $\dfrac{16}{x}$ as $\dfrac{16}{1}$ instead of $-16x^{-2}$.
• Using the gradient of the tangent as the gradient of the normal.
• Assuming every stationary point is a maximum or minimum.

Test yourself on this topic with our free practice questions.