Differentiate products and composites

Identify the outer and inner operations

First identify the structure: sum, product, quotient or composition. Choose the rule from that structure, then differentiate each required factor. A composition also needs the derivative of its inner function.

The basic rules are $d(a)/dx=0$, $d(x^n)/dx=nx^{n-1}$, $d(af)/dx=af^{\prime}$ and $d(f\pm g)/dx=f^{\prime}\pm g^{\prime}$, on intervals where the real functions are differentiable.

A general real exponent does not make $x^n$ real and differentiable at every real x.

For $y=f(u)$ and $u=g(x)$, $dy/dx=(dy/du)(du/dx)$.

Work from the outermost operation inward.

For products use $(uv)^{\prime}=uv^{\prime}+vu^{\prime}$.

For quotients use $(u/v)^{\prime}=(vu^{\prime}-uv^{\prime})/v^2$, with $v\ne0$.

When simplifying a product-rule answer, the source recommends factoring the smaller power; for its rational example it recommends factoring $2x^2$ before expansion.