Question

Using differential, find the approximate value of the $${\left(82\right)}^{\frac{1}{4}}$$ ?

Answer 1

$$\text{ Consider the function }y=f\left(x\right)={\left(x\right)}^{\frac{1}{4}}.$$

$$\text{ Let }:$$

$$x=81$$

$$x+∆x=82$$

$$\text{ Then},$$

$$∆x=1$$

$$\text{ For }x=81,$$

$$y={\left(81\right)}^{\frac{1}{4}}=3$$

$$\text{ Let }:$$

$$dx=∆x=1$$

$$\text{ Now },y={\left(x\right)}^{\frac{1}{4}}$$

$$\Rightarrow \frac{dy}{dx}=\frac{1}{4{\left(x\right)}^{\frac{3}{4}}}$$

$$\Rightarrow {\left(\frac{dy}{dx}\right)}_{x=81}=\frac{1}{108}$$

$$\therefore ∆y=dy=\frac{dy}{dx}dx=\frac{1}{108}\times 1=0.009259$$

$$\Rightarrow ∆y=0.009259$$

$$\therefore {\left(82\right)}^{\frac{1}{4}}=y+∆y=3.009259$$