Question

How do you calculate the derivative of $y=\sqrt{4{{x}^{3}}}$?

Answer 1

This type of question is based on the concept of differentiation. Using the properties of roots, that is $\sqrt{ab}=\sqrt{a}\sqrt{b}$, we get $y=2\sqrt{{{x}^{3}}}$. Then, make some necessary calculations in the given function. Use the power rule of differentiation that is $\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$ and solve. Here, $n=\dfrac{3}{2}$. Cancel out the common term 2 from the numerator and denominator. Do necessary calculations and find the derivative.

Complete step by step solution:
According to the question, we are asked to find the derivative of $y=\sqrt{4{{x}^{3}}}$.
We have been given the function $y=\sqrt{4{{x}^{3}}}$. --------(1)
Let us now simplify the function (1).
We know that $\sqrt{ab}=\sqrt{a}\sqrt{b}$. Let us use this property to simplify the function (1).
Here, a=4 and b=${{x}^{3}}$.
Therefore, we get
$y=\sqrt{4}\sqrt{{{x}^{3}}}$
We know that the square root of 4 is equal to 2, that is $\sqrt{4}=2$. On substituting in the above expression, we get
$\Rightarrow y=2\sqrt{{{x}^{3}}}$
We know that $\sqrt{a}={{a}^{\dfrac{1}{2}}}$. Using this property, we get
$y=2{{x}^{3\times \dfrac{1}{2}}}$
On further simplification, we get
$y=2{{x}^{\dfrac{3}{2}}}$
Now, let us differentiate the function y with respect to x.
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}\left( 2{{x}^{\dfrac{3}{2}}} \right)$
On taking out the constant 2 multiplied with the variable, we get
$\dfrac{dy}{dx}=2\dfrac{d}{dx}\left( {{x}^{\dfrac{3}{2}}} \right)$
Let us use the power rule of differentiation $\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$ to solve this.
Here, $n=\dfrac{3}{2}$.
$\Rightarrow \dfrac{dy}{dx}=2\times \dfrac{3}{2}{{x}^{\dfrac{3}{2}-1}}$
On taking LCM, we get
$\dfrac{dy}{dx}=2\times \dfrac{3}{2}{{x}^{\dfrac{3-2}{2}}}$
On further simplification, we get
$\dfrac{dy}{dx}=2\times \dfrac{3}{2}{{x}^{\dfrac{1}{2}}}$
We find that 2 are common in the numerator and denominator.
On cancelling 2 from the numerator and denominator, we get
$\dfrac{dy}{dx}=3{{x}^{\dfrac{1}{2}}}$
We can express ${{x}^{\dfrac{1}{2}}}$ as $\sqrt{x}$.
Therefore, we get $\dfrac{dy}{dx}=3\sqrt{x}$.
Hence, the differentiation of $y=\sqrt{4{{x}^{3}}}$ is $3\sqrt{x}$.

Note: We should first simplify the given function and then find the derivative. We should avoid calculation mistakes based on sign convention. Use all the identities and rules of differentiation to solve this question. We should differentiate with respect to x and not with respect to y.