Application of Derivatives

Question

Use differentials, find the approximate value of each of the following upto 3 places of decimal:
$left parenthesis 31.9 right parenthesis to the power of 1 fifth end exponent$

Solution

$Take space straight y space equals space straight x to the power of 1 fifth end exponent comma space space space space straight x space equals space 32 comma space space space dx space equals space δx space equals space 0.1 space space space space so space that space straight x space plus space δx space equals space 31.9 Now comma space space space space space straight y space plus space δy space equals space left parenthesis straight x plus δx right parenthesis to the power of 1 fifth end exponent space space space space space rightwards double arrow space space space space space δy space equals space left parenthesis straight x plus space δx right parenthesis to the power of 1 fifth end exponent space minus space straight y space equals space left parenthesis 31.9 right parenthesis to the power of 1 fifth end exponent space minus space 2 rightwards double arrow space space space space left parenthesis 31.9 right parenthesis to the power of 1 fifth end exponent space equals space δy plus 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis$
Now $δy$ is approximately equal to dy
$and space dy space equals space dy over dx dx space equals space 1 fifth straight x to the power of 4 over 5 end exponent dx space equals space fraction numerator 1 over denominator 5 straight x to the power of begin display style 4 over 5 end style end exponent end fraction dx$