Question

How do you find the derivative of ${{e}^{{{x}^{2}}}}$?

Answer 1

To solve the given question, we should know the derivatives of some of the functions, and how to differentiate composite functions. The functions whose derivatives we should know are ${{e}^{x}}$ and ${{x}^{2}}$, their derivatives are ${{e}^{x}}\And 2x$ respectively. The composite functions are functions of the form $f\left( g(x) \right)$, their derivative is found as, $\dfrac{d\left( f\left( g(x) \right) \right)}{dx}=\dfrac{d\left( f\left( g(x) \right) \right)}{d\left( g(x) \right)}\dfrac{d\left( g(x) \right)}{dx}$. We will use these to find the derivative of the given function.

Complete step-by-step answer:
We know that the derivative of the composite function is evaluated as $\dfrac{d\left( f\left( g(x) \right) \right)}{dx}=\dfrac{d\left( f\left( g(x) \right) \right)}{d\left( g(x) \right)}\dfrac{d\left( g(x) \right)}{dx}$.
We are given the function ${{e}^{{{x}^{2}}}}$, we are asked to find its derivative. This is a composite function of the form $f\left( g(x) \right)$, here we have $f(x)={{e}^{x}}\And g(x)={{x}^{2}}$.
To find the derivative of the given function, we need to find $\dfrac{d\left( {{e}^{{{x}^{2}}}} \right)}{d\left( {{x}^{2}} \right)}$, and $\dfrac{d\left( {{x}^{2}} \right)}{dx}$.
We know that the derivative of ${{e}^{x}}$ with respect to x is ${{e}^{x}}$ itself. Thus, the derivative of ${{e}^{{{x}^{2}}}}$ with respect to ${{x}^{2}}$ must be equal to ${{e}^{{{x}^{2}}}}$. Hence, we get $\dfrac{d\left( {{e}^{{{x}^{2}}}} \right)}{d\left( {{x}^{2}} \right)}={{e}^{{{x}^{2}}}}$. Also, the derivative of ${{x}^{2}}$ with respect to x is $2x$.
$\dfrac{d\left( {{e}^{{{x}^{2}}}} \right)}{dx}=\dfrac{d\left( {{e}^{{{x}^{2}}}} \right)}{d\left( {{x}^{2}} \right)}\dfrac{d\left( {{x}^{2}} \right)}{dx}$
Substituting the expressions for the derivative, we get
$\Rightarrow \dfrac{d\left( {{e}^{{{x}^{2}}}} \right)}{dx}={{e}^{{{x}^{2}}}}\times 2x=2x{{e}^{{{x}^{2}}}}$
Thus, the derivative of the given function is $2x{{e}^{{{x}^{2}}}}$.

Note: Here we can express the given function in the form ${{e}^{f(x)}}$. There is a special method to find the derivatives of these types of functions. We can find their derivative using the following method,
$\dfrac{d\left( {{e}^{f(x)}} \right)}{dx}={{e}^{f(x)}}\dfrac{d\left( f(x) \right)}{dx}$
For this question, we have $f(x)={{x}^{2}}$. As we know that the derivative of ${{x}^{2}}$ with respect to x is $2x$. Using the above formula, we get
$\Rightarrow \dfrac{d\left( {{e}^{{{x}^{2}}}} \right)}{dx}={{e}^{{{x}^{2}}}}(2x)=2x{{e}^{{{x}^{2}}}}$
Similarly, we can find the other functions of these types also.