Differentiating Powers of x (DP IB Applications & Interpretation (AI)): Revision Note

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Paul

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4 June 2025

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Differentiating Powers of x

What is differentiation?

Differentiation is the process of finding an expression of the derivative (gradient function) from the expression of a function

How do I differentiate powers of x?

Powers of $x$ are differentiated according to the following formula:
- If $f (x) = x^{n}$ then $f' (x) = n x^{n - 1}$ where $n \in ℤ$
- This is given in the formula booklet
If the power of $x$ is multiplied by a constant then the derivative is also multiplied by that constant
- If $f (x) = a x^{n}$ then $f' (x) = a n x^{n - 1}$ where $n \in ℤ$ and $a$ is a constant
The alternative notation (to $f' (x)$ ) is to use $\frac{d y}{d x}$
- If $y = a x^{n}$ then $\frac{d y}{d x} = a n x^{n - 1}$
  - e.g. If $y = - 4 x^{5}$ then $\frac{d y}{d x} = - 4 \times 5 \times x^{5 - 1} = - 20 x^{4}$
Don't forget these two special cases:
- If $f (x) = a x$ then $f' (x) = a$
  - e.g. If $y = 6 x$ then $\frac{d y}{d x} = 6$
- If $f (x) = a$ then $f' (x) = 0$
  - e.g. If $y = 5$ then $\frac{d y}{d x} = 0$
- These allow you to differentiate linear terms in $x$ and constants
Functions involving fractions with denominators in terms of $x$ will need to be rewritten as negative powers of $x$ first
- e.g. If $f (x) = \frac{4}{x}$ then rewrite as $f (x) = 4 x^{- 1}$ and differentiate

How do I differentiate sums and differences of powers of x?

The formulae for differentiating powers of $x$ apply to all rational powers so it is possible to differentiate any expression that is a sum or difference of powers of $x$
- e.g. If $f (x) = 5 x^{4} + 2 x^{3} - 3 x + 4$ then
  $f' (x) = 5 \times 4 x^{4 - 1} + 2 \times 3 x^{3 - 1} - 3 + 0$
  $f' (x) = 20 x^{3} + 6 x^{2} - 3$
Products and quotients cannot be differentiated in this way so would need expanding/simplifying first
- e.g. If $f (x) = (2 x - 3) (x^{2} - 4)$ then expand to $f (x) = 2 x^{3} - 3 x^{2} - 8 x + 12$ which is a sum/difference of powers of $x$ and can be differentiated

Examiner Tips and Tricks

A common mistake is not simplifying expressions before differentiating
- The derivative of $(x^{2} + 3) (x^{3} - 2 x + 1)$ can not be found by multiplying the derivatives of $(x^{2} + 3)$ and $(x^{3} - 2 x + 1)$

Worked Example

The function $f (x)$ is given by

$f (x) = x^{3} - 2 x^{2} + 3 - \frac{4}{x^{3}}$ , where $x > 0$

Find the derivative of $f (x)$

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