The Chain Rule

The Chain Rule - Exercise 1

Exercise 1. Find the derivative of

$\begin{displaymath}f(t) = \left(t^2 - \frac{2}{t^3} \right)^2\;.\end{displaymath}$

Answer. We will use the Chain rule. Set

$\begin{displaymath}u = t^2 - \frac{2}{t^3}\;\;\mbox{and}\;\; y = f(t)\;.\end{displaymath}$

Then we have y = u². The Chain rule implies

$\begin{displaymath}\frac{dy}{dt} = \frac{du}{dt} \frac{dy}{du}\;.\end{displaymath}$

Since

$\begin{displaymath}\frac{du}{dt} = 2t + \frac{6}{t^4}\;\;\mbox{and}\;\; \frac{dy}{du} = 2u\end{displaymath}$

we get

$\begin{displaymath}\frac{dy}{dt} = \left(2t + \frac{6}{t^4} \right) 2u = 2 \left... ...+ \frac{6}{t^4} \right) \left(t^2 - \frac{2}{t^3} \right) \cdot\end{displaymath}$

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