BolkowN

2021-10-13

Use Version 2 of the Chain Rule to calculate the derivatives of the following functions.
$y={e}^{\mathrm{tan}t}$

svartmaleJ

Step 1
Given function:
$y={e}^{\mathrm{tan}t}$
Differentiate the above function w.r.t 't'
$\frac{dy}{dt}=\frac{d}{dt}{e}^{\mathrm{tan}t}$
The function $y={e}^{\mathrm{tan}t}$ is the composition $f\left(g\left(t\right)\right)$ of two functions .
Step 2
Apply the chain rule
$\frac{d}{dt}\left[f\left(g\left(t\right)\right)\right]=\frac{d}{du}f\left(u\right)\frac{d}{dt}g\left(t\right)$
Here,
So, we get
$\frac{dy}{dt}=\frac{d}{du}{e}^{u}\frac{d}{dt}\mathrm{tan}t$
$\frac{dy}{dt}={e}^{u}×{\mathrm{sec}}^{2}t$
Substitute the value of $u=\mathrm{tan}\left(t\right)$, we get
$\frac{dy}{dt}={e}^{\mathrm{tan}t}×{\mathrm{sec}}^{2}t$

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