How do you differentiate ${e^{2{x^2} + x}}$ using the chain rule?

Hint: Given an exponential expression and we have to differentiate the expression with respect to variable $x$. We can solve the expression by applying the chain rule in which first the whole expression is differentiated with respect to $x$, then the exponent part is differentiated with respect to $x$. The chain rule is applied when the expression contains two or more functions. In the exponential function, the exponent of the expression is also considered as a function.

Formula used:
$\dfrac{d}{{dx}}{e^{ax}} = \dfrac{d}{{dx}}{e^{ax}} \cdot \dfrac{d}{{dx}}ax$
The power rule of the differentiation is defined as:
$\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}$

Complete step-by-step answer:
We are given the exponential expression ${e^{2{x^2} + x}}$. The derivative of the exponential function $\dfrac{d}{{dx}}{e^x} = {e^x}$. First, we will find the derivative of the exponential function using the chain rule by substituting $a = 2{x^2} + x$.
$\dfrac{d}{{dx}}{e^{2{x^2} + x}} = {e^{2{x^2} + x}} \cdot \dfrac{d}{{dx}}\left( {2{x^2} + x} \right)$
Now we will find the derivative of the exponent of the expression using the power rule of the differentiation.
$ \Rightarrow \dfrac{d}{{dx}}2{x^2} + x = 2 \times 2{x^{2 - 1}} + {x^{1 - 1}}$
On simplifying the equation, we get:
$ \Rightarrow 4{x^1} + {x^0}$
Any number raised to the power zero is always equal to one. On simplifying further we get:
$ \Rightarrow 4x + 1$
Then we will multiply the results of both derivatives to get the derivative of the expression.
$ \Rightarrow \dfrac{d}{{dx}}{e^{2{x^2} + x}} = {e^{2{x^2} + x}}\left( {4x + 1} \right)$

Final answer: Hence the derivative of the expression ${e^{2{x^2} + x}}$using the chain rule is ${e^{2{x^2} + x}}\left( {4x + 1} \right)$

Note:
In such types of questions the students mainly don't get an approach on how to solve it. In such types of questions students mainly forget to apply the differentiation on the exponent part separately. In such types of questions, the chain rule is applied which is generally used in composite functions that contain two or more functions.