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Last updated date: 13th Jun 2024
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Hint: In this question, we are given an expression and we have to find its derivative. Take out the constant and find the derivative of only the variable part. Convert the square root of the variable into power and then, use the formula of derivative to find the answer.

Formula used: $\dfrac{{d({x^n})}}{{dx}} = n{x^{n - 1}}$

Complete step-by-step solution:
We are given this expression $2\sqrt x $, and we have to find its derivative.
$ \Rightarrow 2\sqrt x $
Differentiating with respect to x,
$ \Rightarrow \dfrac{{d\left( {2\sqrt x } \right)}}{{dx}}$
Now, we need not apply chain rule here as we only have a constant multiplied with the variable. We will take the constant out.
$ \Rightarrow 2\dfrac{{d\left( {\sqrt x } \right)}}{{dx}}$
Now, we only have to find the derivative of $\sqrt x $. To find that, we will convert the square root into power first.
$ \Rightarrow 2\dfrac{{d\left( {{x^{\dfrac{1}{2}}}} \right)}}{{dx}}$
Now, we will use the formula $\dfrac{{d({x^n})}}{{dx}} = n{x^{n - 1}}$ to find the required derivative by assuming $n = \dfrac{1}{2}$ .
$ \Rightarrow 2 \times \dfrac{1}{2}{x^{\dfrac{1}{2} - 1}}$
Now, on simplifying, we will get,
$ \Rightarrow {x^{ - \dfrac{1}{2}}}$
On rewriting we get,
$ \Rightarrow \dfrac{1}{{\sqrt x }}$

Hence, our answer is $\dfrac{1}{{\sqrt x }}$

Note: Here we used certain rules of powers which were not explained above. Let us look at them below:
The one that we used in the end is - ${x^{ - a}} = \dfrac{1}{{{x^a}}}$. For example: ${x^{ - 4}} = \dfrac{1}{{{x^4}}}$. This rule is called the negative exponent rule.
There are many other rules of powers. They are as follows:
1) Product rule: ${x^a} \times {x^b} = {x^{a + b}}$
2) Quotient rule: $\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$
3) Power rule: ${\left( {{x^a}} \right)^b} = {x^{ab}}$
4) Zero rules: ${x^0} = 1$
There is another rule which says that any number raised to the power “one” equals itself and another rule related to one is that – the number “one” raised to power any number gives us “one” itself. These rules are very handy and help to solve questions easily.