Question

Find the derivative of $\frac{1}{\sqrt{x}}$

Answer 1

Find the derivative of $\frac{1}{\sqrt{x}}$

Let $y=\frac{1}{\sqrt{x}}$

Since, $\frac{1}{\sqrt{x}}={\left(x\right)}^{-\frac{1}{2}}$

Therefore, $y={\left(x\right)}^{-\frac{1}{2}}$

$$\begin{gathered} \Rightarrow \frac{d\left(y\right)}{dx}=\frac{d\left(x^{-{\displaystyle \frac{1}{2}}}\right)}{dx} \\ =-\frac{1}{2}{\left(x\right)}^{-\frac{1}{2}-1}\mathbf{[}\mathbf{\because }\frac{\mathbf{d}\mathbf{\left(}{\mathbf{x}}^{\mathbf{n}}\mathbf{\right)}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathbf{n}\mathbf{.}{\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{]} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}=-\frac{1}{2}{x}^{-\frac{3}{2}} \\ =-\frac{1}{2\sqrt{x^3}} \end{gathered}$$

Hence, the required answer is $-\frac{1}{2x\sqrt{x}}$