Question

$$ \begin{array}{l}{\text { Multiple Choice Which of the following functions is an }} \\ {\text { antiderivative of } \frac{1}{\sqrt{x}} ? \quad \mathrm{}}\end{array} $$ $$ (\mathbf{A})-\frac{1}{\sqrt{2 x^{3}}}(\mathbf{B})-\frac{2}{\sqrt{x}} \quad(\mathbf{C}) \frac{\sqrt{x}}{2}(\mathbf{D}) \sqrt{x}+5(\mathbf{E}) 2 \sqrt{x}-10 $$

Short answer

The correct answer to the problem is option \(E: 2\sqrt{x} - 10\).

Step-by-step solution

Calculate Derivative of the function in Option A

The derivative of \(-\frac{1}{\sqrt{2 x^{3}}}\) using the power rule is \( \frac{3}{2\sqrt[]{2}x^{5/2}}\) which is not equal to \( \frac{1}{\sqrt{x}}. \) Hence, option A is incorrect.

Calculate Derivative of the function in Option B

The derivative of \(-\frac{2}{\sqrt{x}}\) again using the power rule is \( \frac{1}{\sqrt[]{x}}\), which is not correct. Hence, option B is incorrect.

Calculate Derivative of the function in Option C

The derivative of \(\frac{\sqrt{x}}{2}\) is \( \frac{1}{4\sqrt[]{x}},\) which is again not equal to \( \frac{1}{\sqrt{x}}\). Hence, option C is incorrect.

Calculate Derivative of the function in Option D

The derivative of \( \sqrt{x}+5 \) is \( \frac{1}{2\sqrt[]{x}},\) which is not equal to \( \frac{1}{\sqrt{x}}\). So, option D is also incorrect.

Calculate Derivative of the function in Option E

The derivative of \(2\sqrt{x} - 10\) is \( \frac{1}{\sqrt{x}}\), which matches our original function. Therefore, option E is correct.