Definition: The chain rule differentiates a composite function: the derivative of f of g of x is f prime of g of x times g prime of x.
What is the chain rule?
When one function sits inside another, differentiate the outside while leaving the inside alone, then multiply by the derivative of the inside: (f(g(x)))’ = f'(g(x)) times g'(x).
How do you use it?
Worked example: y = (5x2 – 1)4. The outside is “something to the fourth”, the inside is 5x2 – 1.
y’ = 4(5x2 – 1)3 times 10x = 40x(5x2 – 1)3.
At x = 1, the inside equals 4, so y’ = 40 times 64 = 2560.
In Leibniz form, dy/dx = (dy/du)(du/dx), which reads like fractions cancelling.
What goes wrong?
Forgetting the inner derivative, the factor 10x here. Another slip is differentiating the inside too early and changing it. Mathbot recites: outside first with the inside unchanged, then times the inside’s derivative.