Basic Differentiation (AI HL)
Before you reach for the chain, product, or quotient rule, most differentiation starts here: applying the power rule to a polynomial, or recalling the standard derivative of a trig, exponential, or log function. This page covers both, with worked examples and the errors that quietly cost marks. It's part of the broader Differentiation topic.
19 questions on this sub-topic.
The rules you need
The power rule is introduced at SL5.3 for integer powers, and extended at AHL5.9 to cover rational exponents alongside the standard derivatives of \(\sin x\), \(\cos x\), \(\tan x\), \(e^x\), and \(\ln x\). All of these are in the formula booklet.
Power rule
\[\dfrac{d}{dx}(ax^n)=anx^{n-1}\]
Works for any rational \(n\), including negative and fractional powers - rewrite roots and fractions in index form first, then differentiate term by term.
✓ In the formula bookletStandard derivatives
\[\dfrac{d}{dx}(\sin x)=\cos x,\quad \dfrac{d}{dx}(\cos x)=-\sin x,\quad \dfrac{d}{dx}(e^x)=e^x,\quad \dfrac{d}{dx}(\ln x)=\dfrac{1}{x}\]
These only apply directly when the argument is plain \(x\) - anything more complicated, like \(\sin(3x)\), needs the chain rule as well.
✓ In the formula bookletNeed the full syllabus wording and formula-booklet reference table, or a refresher on using your GDC to check a derivative? See Differentiation.
Worked examples
Differentiate.
(a) \(y=3x^4-2x^2+7\)
(b) \(y=5x^{-2}+\sqrt{x}\)
Worked solution
(a) \(\dfrac{dy}{dx} = 12x^3 - 4x.\) M1 A1
(b) \(y = 5x^{-2} + x^{1/2} \Rightarrow \dfrac{dy}{dx}\) M1
\(= -10x^{-3} + \tfrac12 x^{-1/2}.\) A1
Differentiate \(y = e^{2x} + \sin(3x).\)
Worked solution
\(\dfrac{d}{dx}e^{2x}\) M1
\(= 2e^{2x}\) A1
\(\dfrac{d}{dx}\sin(3x)\) M1
\(= 3\cos(3x)\) A1
Differentiate \(y=\ln\!\left(\dfrac{x}{x+1}\right)\), simplifying first.
Worked solution
\(y = \ln x - \ln(x+1).\) M1
\(\dfrac{dy}{dx} = \dfrac1x - \dfrac{1}{x+1}\) M1 A1
\(= \dfrac{1}{x(x+1)}.\) A1
Common mistakes
- Not converting roots and fractions to index form first. \(\sqrt{x}\) and \(\dfrac{2}{x^2}\) need to become \(x^{1/2}\) and \(2x^{-2}\) before the power rule can be applied - trying to differentiate the original form directly leads to errors.
- Dropping the negative sign in the power rule. Differentiating \(5x^{-2}\) gives \(-10x^{-3}\), not \(10x^{-3}\) - the exponent itself is negative, so \(n-1\) makes it more negative still.
- Treating a constant term as if it had a derivative. The constant \(7\) in \(y=3x^4-2x^2+7\) differentiates to \(0\) - a constant on its own has no \(x\) to vary, so it simply disappears.
Ready to practise properly?
19 basic differentiation questions, marked instantly like the real exam.
Quick answers
What is the power rule for differentiation?
If \(f(x) = ax^n\), then \(f'(x) = anx^{n-1}\). This holds for any rational exponent \(n\), including negative and fractional powers, and you differentiate a sum of such terms one term at a time.
What are the standard derivatives I need to know at HL?
The derivatives of \(\sin x\), \(\cos x\), \(\tan x\), \(e^x\), \(\ln x\), and \(x^n\) for rational \(n\). These are all given in the formula booklet, and every other differentiation technique builds on applying them correctly.