Radford Mathematics IB Mathematics resources • AA SL

IB Mathematics: Analysis and Approaches SL — Topic 5 Calculus

Differentiation Rules

Differentiate standard functions and combinations using chain, product and quotient rules.

AA SL · SL 5.6

Learning goal

Select and apply the appropriate differentiation rule with clear notation.

Syllabus link

AA SL: SL 5.6 · Topic 5 Calculus.

Big idea

Most differentiation errors are rule-selection errors. Identify the structure first, then differentiate.

Key relationship

\((uv)\'=u\'v+uv\'\)

1

Core idea

Most differentiation errors are rule-selection errors. Identify the structure first, then differentiate.

Select and apply the appropriate differentiation rule with clear notation.

2

Key results and method

\[(uv)\'=u\'v+uv\'\]
\[\left(\frac{u}{v}\right)\'=\frac{u\'v-uv\'}{v^2}\]
\[\frac{d}{dx}f(g(x))=f\'(g(x))g\'(x)\]
\[(e^x)\'=e^x,\;(\ln x)\'=1/x\]
  1. Rewrite powers/roots if this makes the structure clearer.
  2. Identify whether the expression is a sum, composition, product or quotient.
  3. Apply the rule before simplifying aggressively.
  4. Check that every factor of a chain rule has been included.
3

Worked example

Question

Differentiate \(y=(x^2+1)^5\).

Solution
  1. Outer function: \(u^5\); inner function: \(u=x^2+1\).
  2. \(dy/dx=5(x^2+1)^4(2x)\).
Answer: \(\frac{dy}{dx}=10x(x^2+1)^4\).
4

Practice

Try these questions before moving on.

  1. Differentiate \(x^2e^x\).
  2. Differentiate \(\frac{\sin x}{x}\).
  3. Differentiate \(\ln(3x^2+1)\).

Search the worksheet library for more practice →

5

Tutorials and premium resources

▶ Sum/difference and constant multiple rules
Premium printable lesson notes

The finalized printable handout for this topic is a premium Radford Mathematics resource and is not offered as a free website download.

Visit the Radford Mathematics Store →