Question

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

Answer 1

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 $e \times e \times e \times e \times e \times e$.