Question

How do you find the derivative of $\dfrac{1}{{\sqrt x }}$ ?

Answer 1

Hint : In order to find the first derivative of the above expression with respect to x Use the reciprocal rule of derivation i.e. ${\left[ {\dfrac{1}{{u(x)}}} \right]^\prime } = \dfrac{{u'(x)}}{{u{{(x)}^2}}}$ ,considering $u\left( x \right) = 1 - x$ to solve the above problem . . Later use the formula of differentiation that is Chain rule to solve further and simplify the result . The reason we are using the chain rule as this function can be written as a composition of two functions . The chain rule states $\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}}$ This means we have to differentiate both functions and multiply them and we can obtain the required answer .

Complete step-by-step answer :
Given a function $\dfrac{1}{{\sqrt x }}$ let it be $f(x)$
$f(x) = \dfrac{1}{{\sqrt x }}$
$y = \dfrac{1}{u}$ and $u = \sqrt x = {x^{\dfrac{1}{2}}}$
On simplification further , we have that $y = u$ and $u = {x^{ - \dfrac{1}{2}}}$
Now we will apply the chain rule according to which $\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}}$ . So we have to differentiate both functions and multiply them . Starting with y –
By the power rule $y' = 1 \times {u^0} = 1$ . We have to find the first derivative of the above equation considering it as u : Again by the power rule we get : -
$u' = - \dfrac{1}{2} \times {x^{ - \dfrac{1}{2} - 1}} \\\ \Rightarrow u' = - \dfrac{1}{2}{x^{ - \dfrac{3}{2}}} \\\ \Rightarrow u' = - \dfrac{1}{{2\sqrt {{x^3}} }} \\\ \Rightarrow f'(x) = y' \times u' \\\ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}} \\\ \Rightarrow f'(x) = 1 \times - \dfrac{1}{{2\sqrt {{x^3}} }} \\\ \Rightarrow f'(x) = - \dfrac{1}{{2\sqrt {{x^3}} }} \;$
This $f'(x) = - \dfrac{1}{{2\sqrt {{x^3}} }}$ is our required answer .
So, the correct answer is $f'(x) = - \dfrac{1}{{2\sqrt {{x^3}} }}$.

Note : 1. Calculus consists of two important concepts one is differentiation and other is integration.
2.What is Differentiation?
It is a method by which we can find the derivative of the function .It is a process through which we can find the instantaneous rate of change in a function based on one of its variables.
Let y = f(x) be a function of x. So the rate of change of $y$ per unit change in $x$ is given by:
$\dfrac{{dy}}{{dx}}$.
3. Indefinite integral=Let $f(x)$ be a function .Then the family of all its primitives (or antiderivatives) is called the indefinite integral of $f(x)$ and is denoted by $\int {f(x)} dx$
The symbol $\int {f(x)dx}$ is read as the indefinite integral of $f(x)$ with respect to x.