Key takeaway: The chain rule says the derivative of a composite function f(g(x)) is f′(g(x)) times g′(x): the outer derivative evaluated at the inside, multiplied by the derivative of the inside.
Grade 12 – AP Calculus ABAP Calculus AB Unit 3Calculus
Definition of Chain Rule
The chain rule differentiates a function placed inside another function, such as \((x^3+2x)^6\) or \(\sqrt{5x+1}\). A change in \(x\) first changes the inside quantity, and that change then changes the outside quantity, so two rates multiply.
It works like gears: if one gear turns 3 times for each turn of the first, and the next turns 4 times for each turn of the second, the last turns 12 times per turn of the first.
Formula and notation
With an inside \(u=g(x)\) and an outside \(y=f(u)\):
\[ \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=f^{\prime}(g(x))\,g^{\prime}(x). \]
Differentiate the outside first, leaving the inside untouched, then multiply by the derivative of the inside.
Key facts
- Every layer of nesting adds one more factor, so the chain rule can be applied repeatedly.
- The outer derivative keeps the original inside; only the factor at the end is new.
- Combined with the power rule, it gives the general power rule: n times u to n minus 1 times u-prime.
- It also drives implicit differentiation, where y is treated as a function of x.
Chain Rule example
Differentiate \(y=(x^3+2x)^6\) and find the slope at \(x=1\).
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Outside: power 6. Inside: \(u=x^3+2x\), with \(u^{\prime}=3x^2+2\).
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Combine: \(y^{\prime}=6(x^3+2x)^5\cdot(3x^2+2)\).
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At \(x=1\): the inside equals \(1+2=3\) and \(3x^2+2=5\), so \(y^{\prime}=6\cdot 3^5\cdot 5=6\cdot 243\cdot 5\).
The slope at \(x=1\) is 7,290.
Common mistake
Watch out
A student stops after the outer step and writes \(y^{\prime}=6(x^3+2x)^5\), which gives \(6\cdot 243=1{,}458\) at \(x=1\).
Why it is wrong
That answer skips the derivative of the inside. When the inside is more than a lone \(x\), the factor \(3x^2+2\) must be attached. Only a bare \(x\) has a derivative of 1, which hides the missing factor. The chain rule gives the right slope, 7,290.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus