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How do you find the derivative of $\dfrac{4}{{\sqrt x }}$ ?

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Last updated date: 05th Mar 2024
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IVSAT 2024
Answer
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Hint: To solve these types of questions, directly apply the rules of derivation to the given question. The rule to be applied depends on the form of the function given. After applying the appropriate derivative rule to the given question, simplify the expression obtained to get the final answer.

Formula Used: The following derivative rule can be applied to solve these questions:
If the given function can be written as $f(x) = {x^n}$ then its derivative can be given as $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$

Complete step by step answer:
The given function whose derivative we have to find is $\dfrac{4}{{\sqrt x }}$ .
To find the derivative, first, simplify the given function in terms of exponents of $x$ which can be written as
$\dfrac{4}{{\sqrt x }} = \dfrac{4}{{{x^{1/2}}}}$
The above expression can be further simplified as
$\dfrac{4}{{{x^{1/2}}}} = 4{x^{ - 1/2}}$ (Using the law of exponents: $\dfrac{1}{{{x^n}}} = {x^{ - n}}$ ) $...(i)$
Now compare the above expression to the derivative rule $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$
Since $4$ is a constant term it does not have to undergo derivation.
$\dfrac{d}{{dx}}(4{x^{ - 1/2}}) = 4\dfrac{d}{{dx}}({x^{ - 1/2}})$
Applying the above mentioned derivative rule to the equation $(i)$ we get
$\Rightarrow 4\left( { - \dfrac{1}{2}} \right)({x^{\dfrac{{ - 1}}{2} - 1}})$
Simplifying the above expression we get
$= 4 \times - \dfrac{1}{2}{x^{ - \dfrac{3}{2}}}$
Further dividing the terms to get the simple expression,
$= - 2{x^{ - \dfrac{3}{2}}}$
$= \dfrac{{ - 2}}{{\sqrt {{x^3}} }}$
Hence, the derivative of $\dfrac{4}{{\sqrt x }}$ is $\dfrac{{ - 2}}{{\sqrt {{x^3}} }}$ .

Therefore, $\dfrac{d}{{dx}}(\dfrac{4}{{\sqrt x }}) = \dfrac{{ - 2}}{{\sqrt {{x^3}} }}$.

Additional information:
Differentiation is one of the two important concepts in the field of calculus, apart from integration. It can be defined as a process where we find the instantaneous rate of change in a function based on one of its variables. It is used to find the derivative of a given function. The general expression of derivative of a function $f(x) = y$ can be given as $f'(x) = \dfrac{{dy}}{{dx}}$ where $\dfrac{{dy}}{{dx}}$ can be defined as the rate of change of $y$with respect to $x$ .

Note: While finding the derivative of such questions which include powers of $x$ in the denominator, one should remember the identity $\dfrac{1}{{{x^n}}} = {x^{ - n}}$ which we use to change the negative sign present in the power of the variable $x$ into the positive sign.
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