Question

What is the derivative of \[{{x}^{\dfrac{3}{2}}}\]?

Answer 1

Hint: In this problem we have to find the derivative of \[{{x}^{\dfrac{3}{2}}}\]. Here we can see that we have a power term in the given problem to be differentiated. We should know that, by differentiating a function, we have to change the power or the exponent by 1. Here we can use the differentiating formula \[f'\left( x \right)=n{{x}^{n-1}}\] to find the derivative for the given problem.

Complete step by step answer:
We know that the given problem is,
 \[\Rightarrow {{x}^{\dfrac{3}{2}}}\]
Here we have to differentiate the given problem.
We know that, by differentiating a function, we have to change the power or the exponent by 1.
Here we can use the differentiating formula \[f'\left( x \right)=n{{x}^{n-1}}\] to find the derivative for the given problem.
Where n is \[\dfrac{3}{2}\] as given.
We can now differentiate using the above formula, we get
\[\Rightarrow f'\left( x \right)=\dfrac{3}{2}{{x}^{\dfrac{3}{2}-1}}\]
We can now simplify the above step by reducing the power term, we get
 \[\Rightarrow f'\left( x \right)=\dfrac{3}{2}{{x}^{\dfrac{1}{2}}}\]
Therefore, the derivative of \[{{x}^{\dfrac{3}{2}}}\] is \[f'\left( x \right)=\dfrac{3}{2}{{x}^{\dfrac{1}{2}}}\].

Note: Students should always remember some differentiable formulas, concepts, rules and identities to differentiate these types of problems. We should remember that by differentiating a function, we have to change the power or the exponent by 1. We should also remember that the differentiating formula used here is \[f'\left( x \right)=n{{x}^{n-1}}\], which is used in these types of problems. We should know that differentiating rational power of functions, we can picturize reducing its dimensions by 1.