Chain rule
To differentiate a function inside another function, differentiate the outer one (leaving the inside alone), then multiply by the derivative of the inside.
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Analogy
Gears: if one turns twice as fast as another, and that turns three times as fast as a third, the total ratio is six.
Example
Outer: 5(3x + 1)^4.Inner derivative: 3.Multiply: 15(3x + 1)^4.At x = 0: 15.
In practice
Almost every real function is a composition, so this is the most used rule.
Practice
Flashcards
- What is d/dx of (3x + 1)^5?15(3x + 1)^4, by the chain rule.
Interview questions
- Differentiate f(x) = x^2 sin(3x) and state which rules you used.
- Product rule with u = x^2 and v = sin(3x)
- Power rule: u' = 2x
- Chain rule: v' = 3 cos(3x)
- Result 2x sin(3x) + 3x^2 cos(3x)
Questions people ask
How do I spot when to use the chain rule?
Look for a function inside a bracket, a root, an exponent or a trig function, such as sin of x squared or e to the power of 3x. If the inside is more than just x, differentiate the outside first, then multiply by the inside's derivative.
What is the most common chain rule mistake?
Forgetting to multiply by the derivative of the inside. The derivative of sin(x squared) is cos(x squared) times 2x, not just cos(x squared). A good habit is to circle the inside, differentiate the outside, then always multiply. Writing the inside as a separate letter, such as u = x squared, can also help you keep track of the steps.