Question

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

Answer 1

Hint:Here we have to apply the differentiation to the function. Consider it as y and we differentiate the y with respect to x. The given function can be written as a composite of two functions so we use chain rule to find the derivative. Since it is a trigonometry, we have standard differentiation formulas.

Formula used:
The derivative of \[{e^{ax}}\] is \[\dfrac{d}{{dx}}({e^{ax}}) = {e^{ax}}\dfrac{d}{{dx}}(ax)\]
The derivative of \[a{x^2}\] is \[\dfrac{d}{{dx}}(a{x^2}) = 2ax\]
The chain rule of derivative is \[y = f(x) = g(h(x))\]


Complete step by step answer:
In calculus we have two major topics that are differentiation and integration. We can check if the given function is differentiable by using the limit concept. If the limit exists for the given function then it is differentiable.Now here in this question they have given \[y = {e^{{x^2}}}\].We can say that the given function is composite of two functions.Suppose if we take \[g(x) = {x^2}\] and\[h(x) = {e^x}\]. The composition of function is given as \[g \circ h = h(g(x))\].Therefore we have
\[
g \circ h = h(g(x)) \\
\Rightarrow g \circ h = h({x^2}) \\
\Rightarrow g \circ h = {e^{{x^2}}} \\
\Rightarrow g \circ h = y \\ \]
Since the given function is a composite function of two functions then we can use chain rule of derivative to the given function and hence we can find the derivative of the function.Therefore by applying chain rule of derivative to the function we have
\[
\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{e^{{x^2}}}} \right) \\
\Rightarrow \dfrac{{dy}}{{dx}} = {e^{{x^2}}} \cdot \dfrac{d}{{dx}}\left( {{x^2}} \right) \\ \]
Since the derivative of \[{e^x}\] is \[{e^x}\].
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{{x^2}}}(2x)\]
Since the derivative of \[a{x^2}\] is \[2ax\]
On simplification we have,
\[
\Rightarrow \dfrac{{dy}}{{dx}} = 2x{e^{{x^2}}} \\
\therefore \dfrac{{dy}}{{dx}} = 2xy \\
\]
Hence we obtained the derivative of \[y = {e^{{x^2}}}\] is \[\dfrac{{dy}}{{dx}} = 2xy\].

Note:The differentiation is the rate of change of a function at a point. We must know about the chain rule of derivatives. The function can be written as a composite of two functions, if the function can be written as a composite of two functions then we can apply the chain rule of derivative. Hence by using the derivative formulas we can solve the function and hence obtain the solution.