Differentiating Terms (Positive, Negative & Fractional 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.

Rather than using first principles every time, we can use a general rule for polynomials.

If $f(x) = x^n$ then $f'(x) = nx^{n-1} \quad (n \in \mathbb{R})$

Multiply by the power, reduce the power by 1.

This rule applies to positive, negative, and fractional indices.

Negative indices: negative indices are the most reliable source of dropped marks in differentiation. Reducing the power by 1 makes it more negative — $x^{-2}$ differentiates to $-2x^{-3}$ , not $-2x^{-1}$ .
Rewrite before you differentiate: the power rule only works on terms in the form $ax^n$ . Roots and fractions must be written as indices first.
Consistent lines of working: differentiation is a place where each line must follow logically from the one above. Do not collapse several steps into one line.
Simplify the final answer: an unsimplified derivative can cost the final mark even when the differentiation is perfect.

Example 1

A function $f$ is defined for $x \in \mathbb{R}$ by $f(x) = x^4$ . Find $f'(x)$ .

\begin{aligned} f(x) &= x^4 \\ f'(x) &= 4x^3 \end{aligned}

Example 2

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

\begin{aligned} f(x) &= 2x^3 \\ f'(x) &= 6x^2 \end{aligned}

Example 3

Find $f'(x)$ if $f(x) = x^{-3}$ .

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

Example 4

Find $f'(x)$ if $f(x) = 4x^{-2}$ .

\begin{aligned} f(x) &= 4x^{-2} \\ f'(x) &= -8x^{-3} \end{aligned}

Example 5

Find $f'(x)$ if $f(x) = x^{-\frac{1}{3}}$ .

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