Question

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

Answer 1

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
\[
\begin{aligned}
y &=x^{4}+2+x^{-4} \\
\therefore \frac{\mathrm{d} y}{\mathrm{~d} x} &=4 x^{3}-4 x^{-5}
\end{aligned}
\]