Basic Differentiation Rules (AA HL)
Before the chain, product and quotient rules come into play, every differentiation question rests on a small set of building blocks: the power rule, the standard derivatives of \(\sin x\), \(\cos x\), \(e^x\) and \(\ln x\), and the idea of differentiating from first principles. This page covers those foundations with worked examples. It's part of the broader Differentiation topic.
26 questions on this sub-topic.
The building blocks
Covered under IB syllabus reference SL5.6. Both the power rule and the standard derivatives below are in the formula booklet.
Power rule
\[f(x)=x^n \Rightarrow f'(x)=nx^{n-1}\]
Multiply by the old power, then reduce the power by one. Works for any real \(n\), including negative and fractional powers.
✓ In the formula bookletStandard derivatives
\(\dfrac{d}{dx}(\sin x) = \cos x,\quad \dfrac{d}{dx}(\cos x) = -\sin x\)
\(\dfrac{d}{dx}(e^x) = e^x,\quad \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}\)
These four apply directly, or combine with the sum, chain, product and quotient rules for more complex functions.
✓ In the formula bookletNeed the full syllabus wording and formula-booklet reference table? See Differentiation.
Worked examples
Differentiate \(y = x^4 - 3x^2 + 7.\)
Worked solution
Apply the power rule term by term: M1
\(\dfrac{dy}{dx} = 4x^3 - 6x.\) A1
Use the definition \(f'(x)=\displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\) to find the derivative of \(f(x)=x^2-3x.\)
Worked solution
\(f(x+h) = (x+h)^2 - 3(x+h)\) M1 \(= x^2 + 2xh + h^2 - 3x - 3h.\) A1
\(f(x+h) - f(x) = h(2x + h - 3).\) M1
\(\dfrac{f(x+h)-f(x)}{h} = 2x + h - 3.\) A1
as \(h\to0\), \(f'(x) = 2x - 3.\) A1
Differentiate \(y = 2x^3 + \dfrac{4}{x}.\)
Worked solution
Write \(y = 2x^3 + 4x^{-1}.\) M1
\(\dfrac{dy}{dx} = 6x^2 - 4x^{-2}\) A1
\(= 6x^2 - \dfrac{4}{x^2}.\) A1
Common mistakes
- Misapplying the power rule to negative or fractional powers. \(\dfrac{d}{dx}(x^{-2}) = -2x^{-3}\), not \(-2x^{-1}\) - the power still just drops by one, even when it starts negative or as a fraction.
- Mixing up the sign on \(\cos x\). \(\dfrac{d}{dx}(\sin x) = \cos x\) but \(\dfrac{d}{dx}(\cos x) = -\sin x\) - the minus sign only appears on the second one, and it's easy to drop or add it to the wrong derivative under pressure.
- Rewriting a root or reciprocal incorrectly before differentiating. \(\sqrt{x}\) must become \(x^{1/2}\) and \(\dfrac{1}{x^3}\) must become \(x^{-3}\) before the power rule can be applied - differentiating the un-rewritten form directly is a common source of errors.
Ready to practise properly?
26 basic-differentiation questions, marked instantly like the real exam.
Quick answers
What is the power rule for differentiation?
If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). Multiply by the old power, then reduce the power by one. It applies for any real \(n\), including negative and fractional powers.
What are the standard derivatives I need to know?
\(\dfrac{d}{dx}(\sin x) = \cos x\), \(\dfrac{d}{dx}(\cos x) = -\sin x\), \(\dfrac{d}{dx}(e^x) = e^x\), and \(\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}\). All four are in the formula booklet and combine with the sum, chain, product and quotient rules for more complex functions.
Need a calculator refresher? See Using your GDC on the full Differentiation page.