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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 }}$

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.

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