Preparing to Differentiate 2 – Indices

Video lesson · from 7:125 worked examples

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

Before differentiating, rewrite roots as fractional indices ( $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ ).

Move variables out of the denominator to the numerator using negative indices ( $\frac{1}{x^n} = x^{-n}$ ).

Rewrite roots as fractional indices: $\sqrt{x} = x^{1/2}$ and $\sqrt[3]{x^2} = x^{2/3}$ . You cannot apply the power rule until every term is in the form $ax^n$ .
Subtracting 1 from a fraction: $\tfrac{1}{2} - 1 = -\tfrac{1}{2}$ , so $x^{1/2}$ differentiates to $\tfrac{1}{2}x^{-1/2}$ . This is a very common slip.
Negative indices go more negative: $x^{-3}$ differentiates to $-3x^{-4}$ . Negative indices are the most reliable source of dropped marks in this topic.
Match the form of the question: if it was asked with roots, it is usually best to give the answer with roots too.

Example 1

Differentiate $\sqrt{x}$ with respect to $x$ .

\begin{aligned} y &= x^{\frac{1}{2}} \\ \frac{dy}{dx} &= \frac{1}{2}x^{-\frac{1}{2}} \\ &= \frac{1}{2x^{\frac{1}{2}}} \\ &= \frac{1}{2\sqrt{x}} \end{aligned}

Example 2

Find the derivative of $y = \frac{1}{x^2}$ with respect to $x$ .

\begin{aligned} y &= x^{-2} \\ \frac{dy}{dx} &= -2x^{-3} \\ &= -\frac{2}{x^3} \end{aligned}

Example 3

Differentiate $y = \frac{2}{x}$ with respect to $x$ .

\begin{aligned} y &= 2x^{-1} \\ \frac{dy}{dx} &= -2x^{-2} \\ &= -\frac{2}{x^2} \end{aligned}

Example 4

Find the derivative of $y = \frac{1}{2x}$ with respect to $x$ .

\begin{aligned} y &= \frac{1}{2}x^{-1} \\ \frac{dy}{dx} &= -\frac{1}{2}x^{-2} \\ &= -\frac{1}{2x^2} \end{aligned}

Example 5

Differentiate $\frac{3}{2\sqrt{x}}$ with respect to $x$ .

\begin{aligned} y &= \frac{3}{2x^{\frac{1}{2}}} \\ &= \frac{3}{2}x^{-\frac{1}{2}} \\ \frac{dy}{dx} &= -\frac{1}{2} \cdot \frac{3}{2}x^{-\frac{3}{2}} \\ &= -\frac{3}{4}x^{-\frac{3}{2}} \\ &= -\frac{3}{4\sqrt{x^3}} \end{aligned}