# arrow_back Determine $\frac{\mathrm{d} y}{\mathrm{~d} x}$ if $y=\left(x^{2}+\frac{1}{x^{2}}\right)^{2}$

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Determine $\frac{\mathrm{d} y}{\mathrm{~d} x}$ if
$y=\left(x^{2}+\frac{1}{x^{2}}\right)^{2}$

Multiply out and simplify
We need to get $y$ into a form that we know how to differentiate.
\begin{aligned} y &=\left(x^{2}+\frac{1}{x^{2}}\right)^{2} \\ &=x^{4}+2 \frac{x^{2}}{x^{2}}+\frac{1}{x^{4}} \\ &=x^{4}+2+\frac{1}{x^{4}} \\ &=x^{4}+2+x^{-4} \end{aligned}
Differentiate the simplified expression
\begin{aligned} y &=x^{4}+2+x^{-4} \\ \therefore \frac{\mathrm{d} y}{\mathrm{~d} x} &=4 x^{3}-4 x^{-5} \end{aligned}
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