Question

How do you find the derivative of \[{{e}^{{{x}^{2}}}}\]?

Answer 1

Hint: 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 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 \[f(x)={{e}^{x}}\And g(x)={{x}^{2}}\].
We know that the derivative of the composite function is evaluated as \[\dfrac{d\left( f\left( g(x) \right) \right)}{dx}=\dfrac{d\left( f\left( g(x) \right) \right)}{d\left( g(x) \right)}\dfrac{d\left( g(x) \right)}{dx}\]. 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.