Question

What is the derivative of ${(\sqrt x )^2}$ ?

Answer 1

Hint: The derivative is nothing but the differentiation of a term with respect to another term. The basic formula or method used to calculate the derivative of the given terms is power rule. Power rule is the basic rule of derivative which is used to calculate the derivatives of terms which have power. This is the most simple formula to find the derivative.

Formula used:
Power rule for the derivative such as follow,
${x^n} = n{x^{n - 1}}$ , where n is an integer.

Complete step-by-step answer:
Given,
The term which is needed to differentiate is ${(\sqrt x )^2}$ .
Always first remember to simplify the term which is given to you, because it will reduce more complexity in the problem.
Here first we need to remove the root in the question,
Let us consider the term ${(\sqrt x )^2}$ as $y$ , such as $y = {(\sqrt x )^2}$
We need to find the derivation of $y$ with respect to $x$
The derivative of the $y$ is written as $\dfrac{{dy}}{{dx}}$ .
Now we can simplify the term $y$
To simplify the term, root must be written in terms of power. We know $\sqrt {} $ (square root) is nothing $\dfrac{1}{2}$ , as we substitute this in the term $y$ , we get
$y = {(\sqrt x )^2}$
$y = {({x^{\dfrac{1}{2}}})^2}$
Multiply the term outside the brackets inside the brackets,
$y = {x^{\dfrac{1}{2} \times 2}}$
As we divide the numerator and the denominator in the power, we get
$y = x$
Now we can calculate the derivative of $y$ , because we simplified the term to its smallest power.
We need to use the power rule,
Power rule for the derivative such as follow,
${x^n} = n{x^{n - 1}}$ , where n is an integer.
Buy substituting the value $n = 1$ in power rule we get
$\dfrac{{dy}}{{dx}} = 1 \times {x^{1 - 1}}$
We can subtract the value in the numerator of power, we get
$\dfrac{{dy}}{{dx}} = 1 \times {x^0}$
Any base value to the power $0$ is $1$ , such as ${a^0} = 1$ , we get
$\dfrac{{dy}}{{dx}} = 1 \times 1$
As we multiply the terms we get,
$\dfrac{{dy}}{{dx}} = 1$
Hence we found the derivative of ${(\sqrt x )^2}$ is $1$ .

Note: The derivative of the term should be correctly understood. The power rule of the derivation should be used to simplify the value to reduce the complexity of the sum. Any base value to the power $0$ is $1$ , such as ${a^0} = 1$ , is not ${a^0} \ne 1$ . The root is not negative. Power rule for the derivative such as follow,
${x^n} = n{x^{n - 1}}$ , where n is an integer which is not ${x^n} \ne n{x^{n + 1}}$ .