SL 5.6—Differentiation rules

Product, quotient and chain rules preserve how functions are built.

Differentiate the outer and inner structure rather than expanding blindly: (uv)′=u′v+uv′ and (f(g(x)))′=f′(g)g′.

Example

For y=(x²+1)³, y′=3(x²+1)²·2x=6x(x²+1)².

Name the outer operation and inner function before applying a rule.

The derivative of a product is not the product of derivatives.

Core derivatives: $d(x^n)/dx=nx^{n-1}$ for rational $n$ where defined, $(\sin x)'=\cos x$, $(\cos x)'=-\sin x$, $(e^x)'=e^x$, and $(\ln x)'=1/x$. Also $(u/v)'=(u'v-uv')/v^2$. Example: for $y=x^2e^x$, the product rule gives $y'=e^x(x^2+2x)$; for $y=\ln(3x+1)$, the chain rule gives $3/(3x+1)$.