If f(x)= x-4 2 x , then f'(1) is: - Vidyadip

MCQ

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

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

✓

Answer

Correct option: A.

$\frac{5}{4}$

$\text{f}(\text{x})=\frac{\text{x}-4}{2\sqrt{\text{x}}}$
$=\frac{1}{2}\sqrt{\text{x}}-\frac{2}{\sqrt{\text{x}}}$
$=\frac{1}{2}\text{x}^{\frac{1}{2}}-\text{2x}^{-\frac{1}{2}}$
Differentiate both the sides with respect to $x,$ we get
$\text{f}'(\text{x})=\frac{1}{2}\times\frac{1}{2}\text{x}^{\frac{1}{2}-1}-2\times\Big(-\frac{1}{2}\Big)\text{x}^{-\frac{1}{2}-1}\ [\text{f}(\text{x})=\text{x}^\text{n}$
$\Rightarrow\text{f}'(\text{x})=\text{nx}^{\text{n}-1}]$
$\Rightarrow\text{f}'(\text{x})=\frac{1}{4}\text{x}^{-\frac{1}{2}}+\text{x}^{-\frac{3}{2}}$
$\therefore\text{f}'(\text{1})=\frac{1}{4}\times1+1=\frac{5}{4}$

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