Question

What is the derivative of \[\sqrt {{e^x}} \]?

Answer 1

Hint: In this question, we have to find out the derivative for the given particulars.
We have to differentiate the given function with respect to x. Since it is a composite function we need to use chain rule and applying the differentiation formula, we will get the required result.
Formula:
Chain Rule:
If a function f is a function of g then the derivative of the function \[f(g(x))\] is denoted by \[\dfrac{{df(g(x))}}{{dx}}\] and the chain rule states that: \[\dfrac{{df(g(x))}}{{dx}} = f'(g(x))g'(x)\].
Differentiation formula:
\[\dfrac{{d{x^n}}}{{dx}} = n{x^{n - 1}}\]
\[\dfrac{d}{{dx}}\left( {{e^x}} \right) = {e^x}\]

Complete step-by-step solution:
We need to find out the derivative of \[\sqrt {{e^x}} \],
i.e.\[\dfrac{{d\sqrt {{e^x}} }}{{dx}}\].
Differentiating \[\sqrt {{e^x}} \] using the chain rule which states that:\[\dfrac{{df(g(x))}}{{dx}} = f'(g(x))g'(x)\] where,
\[f(x) = x^{\dfrac{1}{2}},g(x) = {e^x}\]
We get,
\[\dfrac{{d\sqrt {{e^x}} }}{{dx}} = \dfrac{1}{2}{\left( {{e^x}} \right)^{\dfrac{1}{2} - 1}}.\dfrac{{d{e^x}}}{{dx}}\][Using the formula,\[\dfrac{{d{x^n}}}{{dx}} = n{x^{n - 1}}\]]
\[ = \dfrac{1}{2}{\left( {{e^x}} \right)^{ - \dfrac{1}{2}}}.{e^x}\] [Using the formula, \[\dfrac{d}{{dx}}\left( {{e^x}} \right) = {e^x}\]]
\[ = \dfrac{1}{2}{e^x}^{\left( { - \dfrac{1}{2} + 1} \right)}\]
\[ = \dfrac{1}{2}{\left( {{e^x}} \right)^{\dfrac{1}{2}}}\]
\[ = \dfrac{1}{2}\sqrt {{e^x}} \]

Note: The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.
Derivative of a function \[y = f\left( x \right)\] can be written as \[\dfrac{{dy}}{{dx}}\]or \[f'\left( x \right)\].
Composite function:
A composite function is generally a function that is written inside another function. Composition of a function is done by substituting one function into another function. For example, \[f(g(x))\] is the composite function of f $f$ and $g$.
$e$ is the irrational number called the Euler's number. It’s probably the second most commonly known irrational number after $\pi $. Its value is \[2.71828...\] and it goes on.
The $x$ is an exponent indicating how many times to multiply e by itself. The $x$ is a variable. If $x$ is \[2\], that means \[e \times e\] . If x is \[6\], that means \[e \times e \times e \times e \times e \times e\].