Differentiate polynomials: find first and second derivatives, find gradients, tangents and normals, locate and classify stationary points, and solve optimisation problems.

What this dot point is asking

Differentiation is the calculus of change, and CCEA GCSE Further Mathematics introduces it for polynomials. You must differentiate to find the gradient function, find first and second derivatives, use the gradient to write equations of tangents and normals, locate stationary points and classify them as maxima or minima, decide where a function is increasing or decreasing, and solve optimisation problems. This is a major Further Mathematics topic and one of the most heavily examined.

The power rule

Differentiation of a polynomial works term by term using the power rule. Before differentiating, write every term as a power of $x$, including roots and reciprocals.

For example, $y = 4x^3 - 2x^2 + 7x - 5$ differentiates to $\dfrac{dy}{dx} = 12x^2 - 4x + 7$. The derivative is itself a function, so substituting an $x$ value gives the gradient at that point on the curve.

Tangents and normals

The gradient of the curve at a point equals the gradient of the tangent there. The normal is the line perpendicular to the tangent at that point, so its gradient is the negative reciprocal.

To find a tangent: differentiate, substitute the $x$ value to get the gradient, find the $y$ coordinate on the curve, then use $y - y_1 = m(x - x_1)$. For the normal, replace the gradient with its negative reciprocal and use the same point.

Stationary points

A stationary point is where the curve momentarily levels off, so the gradient is zero. Setting the derivative to zero and solving gives their $x$ values; substituting back into the curve gives the coordinates.

Classifying with the second derivative

The second derivative is the derivative of the derivative, and its sign tells you the type of stationary point.

Increasing, decreasing and optimisation

A function is increasing where its derivative is positive and decreasing where its derivative is negative, so you can describe the behaviour of a curve by analysing the sign of $\dfrac{dy}{dx}$. Optimisation problems use this: you write a quantity (such as area or volume) as a function of one variable, differentiate, set the derivative to zero to find the maximum or minimum, and check with the second derivative. These applied questions, for example maximising the volume of an open box, are a classic Further Mathematics challenge and carry several marks.

Why this matters

Differentiation underpins the whole calculus strand and reaches into the Mechanics unit, where differentiating displacement gives velocity and differentiating velocity gives acceleration. The same techniques find the turning points you also meet through completing the square, link to the tangent work of coordinate geometry, and pair with integration as the two halves of calculus. Optimisation in particular rewards a clear method: model, differentiate, solve, classify.