Limits and Derivatives — MATHS STD 11 Science — Question

1. $\frac{5}{4}$

Solution:

$\text{f}(\text{x})=\frac{\text{x}-4}{2\sqrt{\text{x}}}$

$=\frac{1}{2}\sqrt{\text{x}}-\frac{2}{\sqrt{\text{x}}}$

$=\frac{1}{2}\text{x}^{\frac{1}{2}}-\text{2x}^{-\frac{1}{2}}$

Differentiate both the sides with respect to x, we get 

$\text{f}'(\text{x})=\frac{1}{2}\times\frac{1}{2}\text{x}^{\frac{1}{2}-1}-2\times\Big(-\frac{1}{2}\Big)\text{x}^{-\frac{1}{2}-1}\ [\text{f}(\text{x})=\text{x}^\text{n}\Rightarrow\text{f}'(\text{x})=\text{nx}^{\text{n}-1}]$

$\Rightarrow\text{f}'(\text{x})=\frac{1}{4}\text{x}^{-\frac{1}{2}}+\text{x}^{-\frac{3}{2}}$

$\therefore\text{f}'(\text{1})=\frac{1}{4}\times1+1=\frac{5}{4}$