Answer
$y'=(x^{2}+2)^{2}(x^{4}+4)^{4}[\frac{4x}{(x^{2}+2)}+\frac{16x^{3}}{(x^{4}+4)}]$
Work Step by Step
Use logarithmic differentiation to find the derivative of the function.
$y=(x^{2}+2)^{2}(x^{4}+4)^{4}$
Taking the log on both sides.
$lny=ln[(x^{2}+2)^{2}(x^{4}+4)^{4}]$
Use logarithmic properties $ln(xy)=lnx+lny$ and
$ln(x^{y})=ylnx$.
$lny=2ln(x^{2}+2)+4ln(x^{4}+4)$
Differentiate with respect to $x$
$\frac{1}{y}\frac{dy}{dx}=\frac{2}{(x^{2}+2)}\times2x+\frac{4}{(x^{4}+4)}\times4x^{3}$
Hence, $y'=(x^{2}+2)^{2}(x^{4}+4)^{4}[\frac{4x}{(x^{2}+2)}+\frac{16x^{3}}{(x^{4}+4)}]$
