Question

Find the derivative of $\sqrt{2}x+2\sqrt{x}-\frac{1}{\sqrt{x}}$.

• A. $\sqrt{2}+\frac{1}{\sqrt{x}}(1-\frac{1}{2x})$
• B. $\sqrt{2}-\frac{1}{\sqrt{x}}(1+\frac{1}{2x})$
• C. $\sqrt{2}-\frac{1}{\sqrt{x}}(1-\frac{1}{2x})$
• D. $\sqrt{2}+\frac{1}{\sqrt{x}}(1+\frac{1}{2x})$

Answer 1

The correct option is D

$\sqrt{2}+\frac{1}{\sqrt{x}}(1+\frac{1}{2x})$

Finding the derivative:

$$\begin{gathered} y=\sqrt{2}x+2\sqrt{x}-\frac{1}{\sqrt{x}} \\ \frac{dy}{dx}=\sqrt{2}+\frac{2}{2\sqrt{x}}-\frac{(-{\displaystyle \frac{1}{2}})}{x^{{\displaystyle \frac{3}{2}}}} \\ =\sqrt{2}+\frac{1}{\sqrt{x}}+\frac{1}{2x\sqrt{x}} \\ =\sqrt{2}+\frac{1}{\sqrt{x}}(1+\frac{1}{2x}) \end{gathered}$$

Therefore, Option $\left(D\right)$ is correct answer.