The Chain Rule
Introduced| English | Meaning |
|---|---|
| composite function/ˈkɒmpəzɪt ˈfʌŋkʃn/ | A function formed by applying one function to the output of another. |
| Chain Rule/tʃeɪn ruːl/ | The rule for differentiating a composite function by multiplying the appropriate derivatives. |
| outer/ˈaʊtə/ | In a composition f(g(x)), the function f applied to the output of g. |
| inner/ˈɪnə/ | In a composition f(g(x)), the function g evaluated first before its output is passed to f. |
Functions inside functions
- What is the derivative of $(3x+1)^{5}$? It's a power of a function — a composite function 复合函数.
- Expanding it would be brutal. There's a rule built exactly for this: the Chain Rule 链式法则.
- A composite is "an outer function wrapped around an inner one," like $f(g(x))$.
- The Chain Rule differentiates the layers one at a time and multiplies.
The rule
- For $y=f(g(x))$:
-
$$\frac{d}{dx}\big[f(g(x))\big]=f'(g(x))\cdot g'(x)$$
- In words: derivative of the outer (leaving the inner alone) times the derivative of the inner.
- The little "$\cdot\, g'(x)$" is the piece people forget — it's the whole point.
The chain rule: the rates of the outer and inner functions multiply
A composite curve
y = ax² + bx + c
A composite like $(3x+1)^2$ is steeper than its inner line alone — the chain rule multiplies the inner rate in.
The Chain Rule gives $\dfrac{d}{dx}[f(g(x))]=$
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Outer derivative at the inner, times the inner derivative.
Spot the outer and inner
- Before differentiating, name the outer 外层 and inner 内层 functions.
- In $(3x+1)^5$: outer is "(something)$^5$", inner is $3x+1$.
- Outer derivative: $5(3x+1)^4$. Inner derivative: $3$.
- Multiply: $\dfrac{d}{dx}[(3x+1)^5]=5(3x+1)^4\cdot 3=15(3x+1)^4$.
For $y=(2x+1)^3$, find $y'$ at $x=0$. (Chain rule: $3(2x+1)^2\cdot2$.)
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$y'=6(2x+1)^2$; at $0$: $6(1)=6$.
For $\sqrt{x^2+1}=(x^2+1)^{1/2}$, select all correct identifications.
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The square root is the outer; $x^2+1$ is the inner with derivative $2x$.
Chaining with the other rules
- The Chain Rule stacks with everything: power, product, quotient, and elementary derivatives.
- $\dfrac{d}{dx}[\sin(x^2)]=\cos(x^2)\cdot 2x$ — outer $\sin$, inner $x^2$.
- $\dfrac{d}{dx}[e^{3x}]=e^{3x}\cdot 3=3e^{3x}$.
- For deeply nested functions, peel one layer at a time, multiplying each inner derivative as you go.
What is $\dfrac{d}{dx}[\sin(x^2)]$?
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Outer $\cos(x^2)$ times inner derivative $2x$.
The factor most often forgotten in the chain rule is the ____ derivative $g'(x)$.
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You must multiply by $g'(x)$ after differentiating the outer.
What is $\dfrac{d}{dx}[e^{3x}]$?
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Outer $e^{3x}$ times inner derivative $3$.
Never forget to multiply by the inner derivative $g'(x)$. $\frac{d}{dx}[\sin(x^2)]$ is $\cos(x^2)\cdot 2x$, not just $\cos(x^2)$. Leaving off the "$\cdot\,g'(x)$" is the single most common calculus error — the outer derivative alone is only half the answer.
Differentiate $y=\sqrt{x^2+1}$.
- Rewrite: $y=(x^2+1)^{1/2}$. Outer: $(\ )^{1/2}$; inner: $x^2+1$.
- Outer derivative: $\tfrac12(x^2+1)^{-1/2}$. Inner derivative: $2x$.
- $y'=\tfrac12(x^2+1)^{-1/2}\cdot 2x=\dfrac{x}{\sqrt{x^2+1}}$.
The Chain Rule differentiates a composite function: $\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$ — derivative of the outer (inner untouched) times the derivative of the inner. Identify the layers first, and never drop the $\cdot\,g'(x)$ factor. It combines with every other rule for nested expressions.