Calculus that finally makes sense when you hear it explained.

For a composite function $f(g(x))$:

$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

Worked example: differentiate $y = (3x^2 + 1)^5$

• Outer function: $u^5$, derivative $5u^4$
• Inner function: $u = 3x^2+1$, derivative $6x$
• Combine: $\frac{dy}{dx} = 5(3x^2+1)^4 \cdot 6x = 30x(3x^2+1)^4$

How to spot it: if you can describe the function as "something happening to something else," you need the chain rule.