Chain Rule

#math #interview

Chain rule is used to differentiate a variable which is depended on another variable.

For example,
if Hunger = H, Time = t, Crave Ice Cream = Ice and

H = t^{2} + \frac{1}{2}

I c e = s q r t (H)

Then,

\begin{aligned} \frac{\partial H}{\partial t} & = \frac{\partial}{\partial t} (t^{2} + \frac{1}{2}) \\ = 2 t + 0 \\ = 2 t \end{aligned}

And,

\begin{aligned} \frac{\partial I c e}{\partial H} & = \frac{\partial}{\partial H} \sqrt{H} \\ = \frac{1}{2 \sqrt{H}} \end{aligned}

So,

\begin{aligned} \frac{\partial I c e}{\partial t} & = \frac{\partial I c e}{\partial h} * \frac{\partial H}{\partial t} \\ = \frac{1}{2 \sqrt{H}} * 2 t \\ = \frac{t}{\sqrt{H}} \end{aligned}

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