Question

Mark the correct alternative in each of the following:

If $f\left(x\right)=\frac{x-4}{2\sqrt{x}}$, then f '(1) is

(a) $\frac{5}{4}$ (b) $\frac{4}{5}$ (c) 1 (d) 0

Answer 1

$$\begin{gathered} f\left(x\right)=\frac{x-4}{2\sqrt{x}} \\ =\frac{1}{2}\sqrt{x}-\frac{2}{\sqrt{x}} \\ =\frac{1}{2}{x}^{\frac{1}{2}}-2x^{-\frac{1}{2}} \end{gathered}$$

Differentiating both sides with respect to x, we get

$$\begin{gathered} f\text{'}\left(x\right)=\frac{1}{2}\times \frac{1}{2}{x}^{\frac{1}{2}-1}-2\times \left(-\frac{1}{2}\right)x^{-\frac{1}{2}-1}\left[f\left(x\right)=x^n\Rightarrow f\text{'}\left(x\right)=n{x}^{n-1}\right] \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{4}{x}^{-\frac{1}{2}}+x^{-\frac{3}{2}} \\ \therefore f\text{'}\left(1\right)=\frac{1}{4}\times 1+1=\frac{5}{4} \end{gathered}$$

Hence, the correct answer is option (a).