If f(x)=x-4 2 x , then f^ (1) is equal to : - Vidyadip

MCQ

If $f(x)=\frac{x-4}{2 \sqrt{x}}$, then $f^{\prime}(1)$ is equal to :

✓
$\frac{5}{4}$
B
$\frac{4}{5}$
C
1
D
$0$

✓

Answer

Correct option: A.

$\frac{5}{4}$

(A) $\frac{5}{4}$
Explanation : Given, $f(x)=\frac{x-4}{2 \sqrt{x}}$
$f^{\prime}(x)=\frac{1}{2}\left[\frac{\sqrt{x} \cdot 1-(x-4) \cdot \frac{1}{2 \sqrt{x}}}{x}\right]$
$\begin{array}{l}=\frac{1}{2}\left[\frac{2 x-x+4}{2 \sqrt{x} \cdot x}\right] \\ =\frac{1}{2}\left[\frac{x+4}{2 x^{3 / 2}}\right]\end{array}$
Therefore, $\left(f^{\prime}(x)\right)_{\text {at } x=1}=\frac{1}{2}\left[\frac{1+4}{2 \times 1}\right]=\frac{5}{4}$.

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