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Question: Find the derivative of \[\dfrac{1}{{\sqrt x }}\] ....

Find the derivative of 1x\dfrac{1}{{\sqrt x }} .

Explanation

Solution

We know that the given function is in the form of x. The denominator is the main function. We will try to write the form in xn{x^n} because we know the formula to find the derivative of this type. Then after converting into this form we will find the derivative simply.

Formula used:
To find the derivative of the function xn{x^n} we use the formula ddxxn=nxn−1\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}

Complete step-by-step answer :
We are given with a function, 1x\dfrac{1}{{\sqrt x }}
Now we will find the derivative of the function.
For that we can write,
ddx1x\dfrac{d}{{dx}}\dfrac{1}{{\sqrt x }}
But it can also be written as , x=x12\sqrt x = {x^{\dfrac{1}{2}}}
=ddx1x12= \dfrac{d}{{dx}}\dfrac{1}{{{x^{\dfrac{1}{2}}}}}
Now the denominator if written in the numerator it becomes,
=ddxx−12= \dfrac{d}{{dx}}{x^{\dfrac{{ - 1}}{2}}}
Now we can apply the formula mentioned above as n=−12n = \dfrac{{ - 1}}{2}
=−12(x−12−1)= \dfrac{{ - 1}}{2}\left( {{x^{\dfrac{{ - 1}}{2} - 1}}} \right)
Now taking LCM of the power of x we get,
=−12(x−1−22)= \dfrac{{ - 1}}{2}\left( {{x^{\dfrac{{ - 1 - 2}}{2}}}} \right)
On adding the numbers,
=−12(x−32)= \dfrac{{ - 1}}{2}\left( {{x^{\dfrac{{ - 3}}{2}}}} \right)
We can rewrite the bracket in fraction form as,
=−12(1x32)= \dfrac{{ - 1}}{2}\left( {\dfrac{1}{{{x^{\dfrac{3}{2}}}}}} \right)
Thus, ddx1x=−12(1x32)\dfrac{d}{{dx}}\dfrac{1}{{\sqrt x }} = \dfrac{{ - 1}}{2}\left( {\dfrac{1}{{{x^{\dfrac{3}{2}}}}}} \right)
This is the final answer.

Note : In this very easy problem students just because of confusion in the function do make mistakes. As we know, the functions generally used are 1x,xn,x\dfrac{1}{x},{x^n},\sqrt x . The derivative of these is actually very easy but we do make mistakes in writing the solution. Like in the question above the function is having root x but in the fraction form. But we miss it and can unknowingly take the formula for the derivative of either 1xorx\dfrac{1}{x}or\sqrt x . So be careful while using the formula.
Also note that the power of the function is negative so don’t forget to take it with the sign.