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 $${\displaystyle \frac{df(g(x))}{dx}}$$ and the chain rule states that: $${\displaystyle \frac{df(g(x))}{dx}}=f^{\prime }(g(x))g^{\prime }(x)$$.
Differentiation formula:
$${\displaystyle \frac{d{x}^n}{dx}}=n{x}^{n-1}$$
$${\displaystyle \frac{d}{dx}}\left(e^x\right)=e^x$$

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

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 $${\displaystyle \frac{dy}{dx}}$$or $$f^{\prime }\left(x\right)$$.
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 $$e\times e\times e\times e\times e\times e$$.