Compute the derivative $f’(x)$ for $$ f(x)=\log(x^4)\sin(x^3).$$

Solution: By Chain Rule (5.32), we have

$$ \big(\sin(x^3)\big)’=\cos(x^3)(x^3)’=3x^2\cos(x^3). $$

We also have

$$ \big(\log(x^4)\big)’=\big(4\log(x)\big)’=\frac{4}{x}=4x^{-1}. $$

Applying Product Rule (5.29), we obtain

\begin{align*} f’(x)=&\ \log(x^4) \sin(x^3)\\=&\ \big(\log(x^4)\big)’\sin(x^3)+\log(x^4)\big(\sin(x^3)\big)’\\=&\ 4x^{-1}\sin(x^3)+\log(x^4)3x^2\cos(x^3)\\=&\ 4x^{-1}\sin(x^3)+3x^2\log(x^4)\cos(x^3). \end{align*}