The Chain Rule
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Use the chain rule to differentiate \(y=(x^2+2)^3\)
\begin{align}
&y=\left(x^{2}+2\right)^{3} \\
&\begin{aligned}
\text {Let } u &=x^{2}+2 \quad\; y=u^{3} \\
\frac{du}{dx} &=2 x \qquad \frac{dy}{du}=3 u^{2} \\
\frac{d y}{dx} &=\frac{d y}{du} \times \frac{d u}{d x} \\
&=3 u^{2} \times 2 x\\
\frac{d y}{d x} &=6 x \times u^{2} \\
&=6 x\left(x^{2}+2\right)^{2}
\end{aligned}
\end{align}
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