Preparing to Differentiate 1 – Brackets

Video lesson · from 7:122 worked examples

One lesson video covers all of Differentiation, so it opens at 7:12 for this topic — not from the beginning.

Before differentiating, you must rewrite expressions in the form $x^n$ or $ax^n$ .

If the expression contains brackets, you should multiply them out first.

Multiply out completely before differentiating: there is no product rule at Higher, so a product must be expanded first. Differentiating each bracket separately is simply wrong.
Watch signs when expanding: a negative outside a bracket changes every term inside it.
Collect like terms first: tidying up before you differentiate makes the derivative far less error-prone.
Show the expanded form: jumping straight to the derivative loses the working mark, and each line must follow from the one above.

Example 1

A function $f$ is defined for $x \in \mathbb{R}$ by $f(x) = 3x^3 - 2x^2 + 5x$ . Find $f'(x)$ .

Differentiate each term one by one:

\begin{aligned} f(x) &= 3x^3 - 2x^2 + 5x \\ f'(x) &= 9x^2 - 4x + 5 \end{aligned}

Example 2

Find $\frac{dy}{dx}$ when $y = (x - 3)(x + 2)$ .

First multiply out brackets:

\begin{aligned} y &= x^2 + 2x - 3x - 6 \\ y &= x^2 - x - 6 \end{aligned}

Now differentiate:

\frac{dy}{dx} = 2x - 1