The Chain Rule Trap — Derivatives & Calculus (Mathematics)

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Watch Out

Forgetting the Chain Rule

The #1 calculus mistake on CSCA. When the inner function isn't just $x$, you MUST multiply by its derivative. Skip it and the wrong answer is sitting right there to trap you.

(f (g (x)))^{'} = f^{'} (g (x)) \cdot g^{'} (x)

Always multiply by the inner derivative

$(sin x)^{'}$

$= cos x$

Inner is $x$ ; its derivative is 1

$(sin 2 x)^{'}$

$= 2 cos (2 x)$

Inner is $2 x$ ; its derivative is 2

$(sin x^{2})^{'}$

$= 2 x cos (x^{2})$

Inner is $x^{2}$ ; its derivative is $2 x$