#### Provide solution for RD Sharma maths class 12 chapter Differentials Errors and Approximations exercise 13.1 question 9 sub question (xxi)

Answer: $3.00926$

Hint: Here we use below formula,

$\Delta y=f(x+\Delta x)-f(x)$

Given:$(82)^{\frac{1}{4}}$

Solution: $x=81$

$\Delta x=1$

on differentiating  $f(x)$ with respect to  $x$

$\frac{d f}{d x}=\frac{d}{d x}\left(x \frac{1}{4}\right)$

we know  $\frac{d}{d x}\left(x^{3}\right)=n x^{n-1}$

\begin{aligned} &\frac{d f}{d x}=\frac{1}{4} x^{\frac{1}{4}-1} \\\\ &\frac{d f}{d x}=\frac{1}{4} x^{\frac{3}{4}}=\frac{1}{4 x^{\frac{3}{4}}} \end{aligned}

\begin{aligned} &\Rightarrow \text { when } x=81 \text { we have } \frac{d f}{d x}=\frac{1}{4(81)^{\frac{3}{4}}} \\\\ &\Rightarrow \frac{d f}{d x}=\frac{1}{4(34)^{\frac{3}{4}}}=\frac{1}{4\left(3^{3}\right)}=\frac{1}{4 \times 27}=\frac{1}{108}=0.00926 \end{aligned}

\begin{aligned} &\Rightarrow \Delta y=\left(\frac{d y}{d x}\right) \Delta x \\\\ &\Delta f=0.00926 \\\\ &\Rightarrow \text { now, } f(82)=f(81)+\Delta f \end{aligned}

\begin{aligned} &f(82)=3+0.00926 \\\\ &f(82)=3.00926 \end{aligned}