Basic Differentiation Rules (AA SL)
Before you can use the chain, product or quotient rules, you need to be fluent at differentiating single terms - powers of \(x\), roots, and fractions rewritten as powers. This page covers the power rule, the standard derivatives you're expected to know, worked examples, and the mistakes that cost marks. It's part of the broader Differentiation topic.
11 questions on this sub-topic.
The rules you need
Covered under IB syllabus reference SL5.6, which lists the derivatives of \(x^n\) (for \(n \in \mathbb{Q}\)), \(\sin x\), \(\cos x\), \(e^x\) and \(\ln x\), plus differentiation of a sum and a multiple of these functions.
Power rule
\[\dfrac{d}{dx}(ax^n) = anx^{n-1}\]
Multiply by the power, then reduce the power by 1. This works for any rational exponent \(n\), including negative and fractional ones - so roots and fractions must be rewritten as powers first.
Not in the formula booklet - you're expected to know itStandard derivatives
\(\sin x \to \cos x\)
\(\cos x \to -\sin x\)
\(e^x \to e^x\)
\(\ln x \to \dfrac{1}{x}\)
These four are given exactly as shown in the formula booklet - no derivation needed, just recall which one goes with which function.
✓ In the formula bookletNeed the full derivative and integral reference table? See Differentiation.
Worked examples
Find the gradient of \(y = x^2 - 5x + 2\) at \(x = 4\).
Worked solution
The gradient at a point is the derivative there. \(\dfrac{dy}{dx}=2x-5.\) M1
Gradient \(=2(4)-5=3.\) A1
Differentiate \(f(x) = 4\sqrt{x} + \dfrac{3}{x^2}\).
Worked solution
Rewrite \(f(x)\) using negative and fractional powers: \(f(x)=4x^{1/2}+3x^{-2}.\) M1
Differentiate term by term: \(f'(x)=2x^{-1/2}-6x^{-3}.\) A1
Rewrite the derivative without the fractional and negative powers: \(f'(x)=\dfrac{2}{\sqrt x}-\dfrac{6}{x^3}.\) A1
A balloon's volume is \(V(t)=4t^2-t^3\) cm\(^3\) for \(0\le t\le4\).
Use your GDC to find the rate of change of volume at \(t=1\) s.
Worked solution
\(V'(t)=8t-3t^2\)M1
Subsistutite in the value t=1
\(8(1)-3(1)=5\) cm\(^3\)s\(^{-1}\) A1
Common mistakes
- Differentiating a root or fraction without rewriting it as a power first. \(\sqrt{x}\) and \(\dfrac{3}{x^2}\) can't be differentiated directly - convert to \(x^{1/2}\) and \(3x^{-2}\), apply the power rule, then convert back if the question expects roots or fractions in the answer.
- Dropping the minus sign when differentiating \(\cos x\). The derivative of \(\cos x\) is \(-\sin x\), not \(\sin x\) - this sign is easy to lose under time pressure.
- Stopping at the derivative when the question asks for a gradient at a point. \(\dfrac{dy}{dx}=2x-5\) is a formula, not the answer - you still need to substitute the given \(x\)-value and evaluate.
Ready to practise properly?
10 basic differentiation questions, marked instantly like the real exam.
Quick answers
What is the power rule for differentiation?
\(\dfrac{d}{dx}(ax^n) = anx^{n-1}\): multiply by the power, then reduce the power by 1. It works for any rational exponent, including negative and fractional ones.
How do you differentiate a square root or a fraction like \(\frac{1}{x^2}\)?
Rewrite it as a power first: \(\sqrt{x} = x^{1/2}\) and \(\frac{1}{x^2} = x^{-2}\). Then apply the power rule to each term before simplifying back to root or fraction form if needed.