How do you solve $f\left( x \right)={{e}^{\tan x}}$ using the chain rule?

Hint: In this question we have a composite function which has no direct formula for calculating the derivative therefore, we will use the chain rule on the function which is $f'(x)=g'(h(x))h'(x)$.
We will consider in the function the outer function to be $g(x)={{e}^{x}}$ and the inner function $h\left( x \right)=\tan x$ . We will then differentiate the terms and simplify to get the required solution.

Complete step-by-step solution:
We have the equation given as:
$\Rightarrow f\left( x \right)={{e}^{\tan x}}$
Since we have to find the derivative of the term, it can be written as:
$\Rightarrow f'\left( x \right)=\dfrac{d}{dx}{{e}^{\tan x}}$
Now since there is no direct formula for calculating the derivative of the given expression, we will use the chain rule which is $f'(x)=g'(h(x))h'(x)$.
We know that $\dfrac{d}{dx}{{e}^{x}}={{e}^{x}}$, and since we are using chain rule, we will write it as:
$\Rightarrow f'\left( x \right)={{e}^{\tan x}}\dfrac{d}{dx}\tan x$
We know that $\dfrac{d}{dx}\tan x={{\sec }^{2}}x$ therefore, on substituting, we get:
$\Rightarrow f'\left( x \right)={{e}^{\tan x}}\times {{\sec }^{2}}x$
On simplifying the term, we get:
$\Rightarrow f'\left( x \right)={{\sec }^{2}}x{{e}^{\tan x}}$, which is the required solution.

Note: It is to be remembered that the questions which have the property of chain rule and product rule are two entirely different concepts. The chain rule is used when there is a composite function present in the form of $g\left( h\left( x \right) \right)$ and the product rule is to be used when two functions are in multiplication which implies, they are in the form $g\left( x \right)\times h\left( x \right)$. In both the chain rule and the product rule, there can be more than two terms or functions therefore they have to be solved accordingly.