Use Version 2 of the Chain Rule to calculate the derivatives of the following functions....

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