To differentiate composite functions of the form f(g(x)) we use the chain rule (or "function of a function" rule). The derivative of the function of a function f(g(x)) can be expressed as:
f'(g(x)).g'(x)
Alternatively if y=f(u) and u = g(x) then \displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}
Alternatively if y=f(u) and u = g(x) then \displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}
Summary/Background
The chain
rule is also known as the "function of a function" rule. It can be stated in a
number of ways.
If y = f(u), u = g(x) then \displaystyle\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx}
Here are some more examples:
\displaystyle \frac{d((x+1)^6)}{dx} = 6(x+1)^5.1 = 6(x+1)^5
\displaystyle \frac{d(e^{10x})}{dx} = e^{10x}.10 = 10e^{10x}
\displaystyle \frac{d( \ln x^2)}{dx} = \frac{1}{x^2}.2x = \frac{2}{x}
If y = f(u), u = g(x) then \displaystyle\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx}
Here are some more examples:
\displaystyle \frac{d((x+1)^6)}{dx} = 6(x+1)^5.1 = 6(x+1)^5
\displaystyle \frac{d(e^{10x})}{dx} = e^{10x}.10 = 10e^{10x}
\displaystyle \frac{d( \ln x^2)}{dx} = \frac{1}{x^2}.2x = \frac{2}{x}
Software/Applets used on this page
Glossary
chain rule
A rule for differentiating a function of a function:
dy/dx = dy/du x du/dx.
dy/dx = dy/du x du/dx.
composite
made of a combination of simpler shapes or bodies
derivative
rate of change, dy/dx, f'(x), , Dx.
differentiate
to find the derivative of a function
function
A rule that connects one value in one set with one and only one value in another set.
rule
A method for connecting one value with another.
union
The union of two sets A and B is the set containing all the elements of A and B.
This question appears in the following syllabi:
Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|
AP Calculus AB (USA) | 3 | Differentiation | Rules - the chain rule | - |
AP Calculus BC (USA) | 3 | Differentiation | Rules - the chain rule | - |
AQA A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule | - |
AQA A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule | - |
AQA AS/A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule | - |
CBSE XII (India) | Calculus | Continuity and Differentiability | Derivative of composite functions, chain rule | - |
CBSE XI (India) | Calculus | Limits and Derivatives | Sum, difference, product and quotient rules | - |
CCEA A-Level (NI) | C3 | Differentiation | Rules - the chain rule | - |
CIE A-Level (UK) | P2 | Differentiation | Rules - the chain rule | - |
Edexcel A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule | - |
Edexcel A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule | - |
Edexcel AS/A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule | - |
I.B. Higher Level | 6 | Differentiation | Rules - the chain rule | - |
I.B. Standard Level | 6 | Differentiation | Rules - the chain rule | - |
Methods (UK) | M8 | Differentiation | Rules - the chain rule | - |
OCR A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule | - |
OCR A2 Maths 2017 | Pure Maths | Differentiation Techniques | Chain Rule | - |
OCR MEI A2 Maths 2017 | Pure Maths | Differentiation Techniques | Chain Rule | - |
OCR-MEI A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule | - |
Pre-U A-Level (UK) | 4 | Differentiation | Rules - the chain rule | - |
Scottish Advanced Highers | M1 | Differentiation | Rules - the chain rule | - |
Scottish (Highers + Advanced) | HM3 | Differentiation | Rules - the chain rule | - |
Scottish Highers | M3 | Differentiation | Rules - the chain rule | - |
Universal (all site questions) | D | Differentiation | Rules - the chain rule | - |
WJEC A-Level (Wales) | C3 | Differentiation | Rules - the chain rule | - |