Question

How do you find the derivative of $y = \sqrt x $ ?

Answer 1

Hint:
We will find out the derivative of the above term by differentiation, in which the power of a variable will become coefficient and the power of the variable will be subtracted by one and then simplify the terms.

Complete step by step solution:
To find out the derivative of $y = \sqrt x $ first of all we will simplify it as we know, $\sqrt x = {x^{\dfrac{1}{2}}}$
As $y = {x^{\dfrac{1}{2}}}......(1)$
 Now, we will differentiate equation (1) on both sides.
$\dfrac{{dy}}{{dx}} = \dfrac{1}{2}{x^{\dfrac{1}{2} - 1}}$
We will take L.C.M. of the powers of $x$
\[ \Rightarrow \dfrac{1}{2}{x^{\dfrac{{1 - 2}}{2}}}\]
As we observe that we can perform subtraction operation in powers of $x$
$ = \dfrac{1}{2}{x^{\dfrac{{ - 1}}{2}}}$
As we know that, ${a^{ - n}} = \dfrac{1}{{{a^n}}}$ so we can remove negative signs from power by taking it to the denominator.
\[ = \dfrac{1}{2} \times \dfrac{1}{{{x^{\dfrac{1}{2}}}}}\]
As we know that ${x^{\dfrac{1}{2}}} = \sqrt x $
$\dfrac{{dy}}{{dx}} = \dfrac{1}{2} \times \dfrac{1}{{\sqrt x }} \Rightarrow \dfrac{1}{{2\sqrt x }}$

Hence, derivative of $y$ is $\dfrac{1}{{2\sqrt x }}$

Note:
We should know that we can perform derivative by differentiation, and also remember the identity ${a^{ - n}} = \dfrac{1}{{{a^n}}}$ which we use to change the negative sign present in the power of the variable into the positive sign.