Answer
$\displaystyle \frac{dy}{dx}=4(8x^{4}-5x^{2}+1)^{3}\cdot(32x^{3}-10x)$
Work Step by Step
Recognize the function as a composite function:
$y=(8x^{4}-5x^{2}+1)^{4}=[g(x)]^{4},$
$f(x)=x^{4},\qquad g(x)=8x^{4}-5x^{2}+1$
Apply the chain rule.
$\displaystyle \color{blue}{\frac{dy}{dx}=\{f[g(x)]\}^{\prime}=f^{\prime}[g(x)]\cdot g^{\prime}(x).}$
$\left[\begin{array}{lll}
f(x)=x^{4} & , & g(x)=8x^{4}-5x^{2}+1.\\
f^{\prime}(x)=4x^{3} & & g^{\prime}(x)=32x^{3}-10x\\
f^{\prime}[g(x)]=4(8x^{4}-5x^{2}+1)^{3} & &
\end{array}\right]$
$\displaystyle \frac{dy}{dx}=4(8x^{4}-5x^{2}+1)^{3}\cdot(32x^{3}-10x)$
