2.4.1—Advanced differentiation

Differentiate e^{g(x)} as g′(x)e^{g(x)}, ln(g(x)) as g′(x)/g(x), and sin or cos of an inner function using the chain rule.

Identify the outer function first, differentiate it, then multiply by the derivative of the inner expression. Simplify only after all factors are present.

d[ln(1+x²)]/dx=2x/(1+x²), while d[e^{3x}sin x]/dx=e^{3x}(3sin x+cos x).

The derivative of ln g is not 1/ln g, and forgetting g′ changes the scale of the answer.