Preparing to Differentiate 3 – Indices and Fractions

Video lesson · from 7:123 worked examples

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

Theory

If you have a fraction with a single term in the denominator, you should separate it into multiple fractions and simplify the indices before differentiating.

⚠️ Common Examiner Traps

Split the fraction term by term: $\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$ . You can only do this when the denominator is a single term.
You cannot split the other way: $\frac{1}{a+b}$ is not $\frac{1}{a} + \frac{1}{b}$ . A sum in the denominator cannot be separated.
Subtract the indices: $\frac{x^3}{x} = x^2$ . Dividing by $x$ reduces the power by one; it does not divide the coefficient.
Simplify fully before differentiating: every term must be $ax^n$ first. Half-simplified terms lead to wrong powers.

Worked examples

Example 1

A function $f$ is defined for $x \in \mathbb{R}, x \neq 0$ by $f(x) = \frac{x}{5} + \frac{1}{x^2}$ . Find $f'(x)$ .

Prepare the terms:

\begin{aligned} f(x) &= \frac{1}{5}x + x^{-2} \end{aligned}

Differentiate:

\begin{aligned} f'(x) &= \frac{1}{5} - 2x^{-3} \\ &= \frac{1}{5} - \frac{2}{x^3} \end{aligned}

Example 2

Differentiate $\frac{x^4 - 3x^2}{5x}$ with respect to $x$ .

Split into separate fractions and prepare:

\begin{aligned} f(x) &= \frac{x^4}{5x} - \frac{3x^2}{5x} \\ &= \frac{1}{5}x^3 - \frac{3}{5}x \end{aligned}

Now differentiate:

f'(x) = \frac{3}{5}x^2 - \frac{3}{5}

Example 3

Find the derivative of $y = \sqrt{x}(x^2 + \sqrt[3]{x})$ with respect to $x$ .

Write as fractional indices, then expand the brackets:

\begin{aligned} y &= x^{\frac{1}{2}} (x^2 + x^{\frac{1}{3}}) \\ y &= x^{\frac{1}{2}}x^2 + x^{\frac{1}{2}}x^{\frac{1}{3}} \\ y &= x^{\frac{5}{2}} + x^{\frac{5}{6}} \end{aligned}

Remember to add powers when multiplying: $\frac{1}{2} + 2 = \frac{5}{2}$ and $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .

Now differentiate:

\begin{aligned} \frac{dy}{dx} &= \frac{5}{2}x^{\frac{3}{2}} + \frac{5}{6}x^{-\frac{1}{6}} \\ &= \frac{5}{2}\sqrt{x^3} + \frac{5}{6\sqrt[6]{x}} \end{aligned}