Question

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

Answer 1

Hint: For solving this question you should know about the differentiation of normal functions and how to calculate the derivatives. In this question we will differentiate \[{{x}^{{3}/{2}\;}}\] with respect to any variable x. But as we know that the \[{{x}^{{3}/{2}\;}}\] is not constant value and the differentiation of constant is 0 because it will never change so the rate of change constant function will be zero. But here it is constant so here some answers will appear as variables. According to our question, if we see that it is asked to us to determine the derivative of \[{{x}^{{3}/{2}\;}}\].

Complete step-by-step answer:
So, as we know that the differentiation of any exponential will be as \[\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}\].
And we know that the differentiation of the constant is always zero.
And the derivatives of constants are always zero because they do not change with the variable in whose respect they are going to differentiate.
So, the differentiation of \[{{x}^{{3}/{2}\;}}\]:
\[\begin{align}
  & \Rightarrow \dfrac{d}{dx}{{x}^{{3}/{2}\;}}=\dfrac{3}{2}{{x}^{\left( {3}/{2}\;-1 \right)}} \\
 & \Rightarrow \dfrac{d}{dx}{{x}^{{3}/{2}\;}}=\dfrac{3}{2}{{x}^{{1}/{2}\;}} \\
\end{align}\]
So, the derivative of \[{{x}^{{3}/{2}\;}}\] is equal to \[\dfrac{3}{2}{{x}^{{1}/{2}\;}}\].

Note: During solving the differentiation of any term we always have to be assure that the term which we are differentiating and the variable with whose respect we differentiate to this, are always have same variable, unless this will be a constant for that and the differentiation will be zero.