AHL 5.9 (HL)—Further differentiation

Know $(\sin x)'=\cos x$, $(\cos x)'=-\sin x$, $(\tan x)'=\sec^2x$, $(e^x)'=e^x$, $(\ln x)'=1/x$ and $(x^n)'=nx^{n-1}$ for rational $n$ on the appropriate domain.

Use chain $(f(g(x)))'=f'(g(x))g'(x)$, product $(uv)'=u'v+uv'$ and quotient $(u/v)'=(u'v-uv')/v^2$. Identify the outer structure before choosing a rule, and combine rules when functions are nested.

Example

For $y=x^2e^{3x}$, $dy/dx=2xe^{3x}+3x^2e^{3x}=xe^{3x}(2+3x)$. In related rates, differentiate the connecting equation with respect to time, then substitute values with units only after differentiating.

Do not confuse this Objective with second derivatives in AHL 5.10. Product and quotient rules require both terms, and the chain rule requires the derivative of the inner function.