The chain rule
A composite function applies one function to the output of another. Differentiate the outer function with the inner expression kept in place, then multiply by the derivative of the inner function
\[\frac{d}{dx}F\left(g(x)\right)=F'\left(g(x)\right)\times g'(x)\]
- Name the inner function The inner function is applied first and is the expression inside the outer function \(\displaystyle g(x)=3x-2,\quad F(u)=u^{4}\)
- Differentiate the outer function Differentiate the outer rule while keeping its input as a single symbol \(\displaystyle F'(u)=4u^{3}\)
- Differentiate the inner function Differentiate the expression that is applied first \(\displaystyle g'(x)=3\)
- Multiply the factors Evaluate the outer derivative at the inner function and multiply by the inner derivative \(\displaystyle \frac{d}{dx}F\left(g(x)\right)=F'\left(g(x)\right)\times g'(x)\)
Worked example
Differentiate the function and find its derivative at the stated input
\[f(x)=\left(2x+1\right)^{3}\]
- \(\displaystyle g(x)=2x+1,\quad F(u)=u^{3}\)The linear expression is the inner function and the cube is the outer function
- \(\displaystyle F'(u)=3u^{2}\)Differentiate the outer power while keeping u as its input
- \(\displaystyle g'(x)=2\)Differentiate the inner linear function
- \(\displaystyle f'(2)=3\left(2\times2+1\right)^{2}\times2=3\times25\times2=150\)Substitute x = 2 into the inner function and multiply the two derivative factors
Common mistakes
Omitting the inner derivative
✕\(\displaystyle \frac{d}{dx}\left(3x-2\right)^{4}=4\left(3x-2\right)^{3}\)✓\(\displaystyle \frac{d}{dx}\left(3x-2\right)^{4}=4\left(3x-2\right)^{3}\times3\)
The chain rule includes the derivative of the inner function as a multiplication factorEvaluating the outer derivative at x
✕\(\displaystyle \frac{d}{dx}\left(3x-2\right)^{4}=4x^{3}\times3\)✓\(\displaystyle \frac{d}{dx}\left(3x-2\right)^{4}=4\left(3x-2\right)^{3}\times3\)
Evaluate the outer derivative at the inner function, keeping the inner expression inside the powerAdding the derivative factors
✕\(\displaystyle \frac{d}{dx}\left(3x-2\right)^{4}=4\left(3x-2\right)^{3}+3\)✓\(\displaystyle \frac{d}{dx}\left(3x-2\right)^{4}=4\left(3x-2\right)^{3}\times3\)
The chain rule combines the outer and inner derivatives by multiplication, not additionDifficulty levels
introDifferentiate a power of a linear function in factorised form
standardDifferentiate a polynomial inner function or evaluate an exponential or circular composite at an exact input
challengeDifferentiate a two-level composite or form a tangent or normal gradient
examSelect a derivative form and complete an exact coefficient or value check
Questions about this topic
How can the inner function be identified?+
The inner function is the expression inside the outer operation. For a fourth power of a linear expression, the linear expression is the inner function
Why is the inner derivative multiplied?+
The output of the inner function changes as x changes, so its rate of change is a factor in the total rate of change
Where is the outer derivative evaluated?+
It is evaluated at the inner function, not at x by itself. The inner expression remains inside the outer derivative rule
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 6: Applications of differentiation
In this chapter
Differentiate polynomialsChallenge enrichment: rational powers and derivative domainsDifferentiate natural exponential functions and solve simple derivative conditionsDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsDifferentiate a composite function by the chain ruleDifferentiate a product or quotient of two functionsEvaluate and determine function rules6B Differentiation rules: Find the maximal domain of an algebraic functionExpand and factor polynomial expressionsSimplify and combine algebraic fractions6B Differentiation rules: Find the equation of a line from its graph or from its intercept and slopeAnalyse polynomial curves, stationary points and extremaUse a derivative to analyse and recognise the graph of a function6C Tangents, normals and curve sketching: Evaluate a polynomial or recover a coefficient parameter6C Tangents, normals and curve sketching: Move between roots, factors, polynomial rules and equationsRecognise and classify a polynomialConnect a factorised higher-degree rule and its graphAnalyse a cubic graph or determine its ruleWork with sets and interval notation6C Tangents, normals and curve sketching: Find the maximal domain of an algebraic function6C Tangents, normals and curve sketching: Find the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a maximum or minimum problem by differentiationSet up and solve a practical linear equationRearrange literal equations and recognise equivalent line forms6D Optimisation: Find the maximal domain of an algebraic function6D Optimisation: Evaluate a polynomial or recover a coefficient parameter6D Optimisation: Move between roots, factors, polynomial rules and equationsCalculate and interpret average and instantaneous rates from a modelVolume of cones, pyramids and spheresVolume of prisms and cylindersPerimeter and area of standard shapesSimilar figures and scale factorsFind a side of a right-angled triangleAnalyse straight-line motion through displacement, velocity and acceleration6F Kinematics: Evaluate a polynomial or recover a coefficient parameter