Question

What is the derivative of $5\left( \sqrt{x} \right)$ ?

Answer 1

Here in this question we have been asked to find the derivative of $5\sqrt{x}$ for answering this question we will use the following formula from the concepts of derivations given as $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$ and simplify the given expression.

Complete step by step answer:
Now considering the question we have been asked to find the value of the derivative of the given expression $5\sqrt{x}$ .
We can write the given expression mathematically as $\dfrac{d}{dx}5\sqrt{x}$ .
From the basic concepts of derivations we know that the derivative of a constant is a zero and the derivative of product of a constant and a function is product of the derivative of the function and the constant this can be mathematically given as $\dfrac{d}{dx}af\left( x \right)=a\dfrac{d}{dx}f\left( x \right)$ .
Hence we can write the given expression as $5\dfrac{d}{dx}\sqrt{x}$ .
This expression can be further simplified and written as $5\dfrac{d}{dx}{{x}^{\dfrac{1}{2}}}$ .
From the basic concepts of derivations we know that a basic formula can be used in this case which is given as $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$ .
By applying this formula in this case we will get $5\left( \dfrac{1}{2}{{x}^{\dfrac{-1}{2}}} \right)$ .
By further simplifying this expression we will have $\dfrac{5}{2\sqrt{x}}$

Therefore we can conclude that the value of the derivative of the given expression $5\sqrt{x}$ will be given as $\dfrac{5}{2\sqrt{x}}$

Note: In the process of answering questions of this type we should be very sure with our formula that we are using and calculations that we are performing in between the steps. Someone can make a mistake if they consider the given expression as $5\left( \dfrac{1}{2}{{x}^{\dfrac{-1}{2}}} \right)=\dfrac{5}{2}\sqrt{x}$ which leads them to end up having a wrong conclusion.