Question

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

Answer 1

Hint: 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 \[{x^n}\] 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 \[{x^n}\] we use the formula \[\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}\]

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