Chain Rule
In short: Chain rule handles functions nested inside one another: the rate of the whole equals the outer rate, read at the inner value, multiplied by the inner rate.
AP Calculus ABCalculusAP Calculus AB Unit 3
What is Chain Rule?
The chain rule handles a function that is fed into another function, such as \(\sqrt{7x^2 + 2}\) where \(7x^2 + 2\) goes inside a square root. Rates along a chain multiply, like linked bicycle gears: if the middle quantity runs 3 times faster than \(x\) and the output runs 14 times faster than the middle quantity, the output runs 42 times faster than \(x\).
In symbols: \(\frac{d}{dx}\left[f(g(x))\right] = f^{\prime}(g(x)) \cdot g^{\prime}(x)\), or \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\).
How to spot it
- A function sits inside another: a power of a bracket, a root of an expression, or sine of something other than x.
- The expression has parentheses raised to a large exponent, so expanding would take forever.
- You are given y as a function of u and u as a function of x, then asked for dy/dx.
- A related-rates or implicit problem has a variable like y that secretly depends on x.
The method
- Identify the outer function and the inner function, and name the inner expression \(u\).
- Differentiate the outer function while leaving \(u\) untouched inside it.
- Multiply by the derivative of the inner function, \(\frac{du}{dx}\).
- Substitute back for \(u\) and simplify, or evaluate at the requested input.
Chain Rule: solved example
Problem
Find \(h^{\prime}(1)\) for \(h(x) = \sqrt{7x^2 + 2}\).
Solution
Outer: \(\sqrt{u} = u^{1/2}\), with derivative \(\frac{1}{2}u^{-1/2}\). Inner: \(u = 7x^2 + 2\), with derivative \(14x\).
Multiply: \(h^{\prime}(x) = \frac{1}{2\sqrt{7x^2 + 2}} \cdot 14x = \frac{7x}{\sqrt{7x^2 + 2}}\).
At \(x = 1\): \(\sqrt{9} = 3\), so \(h^{\prime}(1) = \frac{7}{3}\).
Try it
Question 1
Differentiate \(y = (2x – 9)^6\).
Show the answer
Outer derivative \(6(2x-9)^5\), times inner derivative 2: \(y^{\prime} = 12(2x – 9)^5\).
Question 2
Differentiate \(y = \sin(5x^2)\).
Show the answer
Outer: \(\cos(5x^2)\). Inner derivative: \(10x\). So \(y^{\prime} = 10x\cos(5x^2)\).
Chain Rule vs. Product Rule
The chain rule applies when one function is nested inside another, like \(\cos(x^3)\). The product rule applies when two functions sit side by side and are multiplied, like \(x^3 \cos x\). Some problems need both, for example \(x^3\cos(x^3)\): product on the outside, chain inside.
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Whole dictionary: A–Z · AP Calculus AB · Calculus