 Hello friends, let's work out the following problem. It says using differentials find the approximate value of the following up to 3 places of the decimal. So let's now move on to the solution and first of all define y as a function of x and let's take it to be x to the power 1 by 4 and let x be 16. Here we choose x in such a way so that we can easily find out its fourth root and we know that we can very easily find out the fourth root of 16 that is 2. So now we take delta x as minus 1 as we need to have x plus delta x as 15 so this would be 16 minus 1 that is 15. Now we know that delta y is equal to f of x plus delta x minus fx so this is x plus delta x to the power 1 by 4 minus x to the power 1 by 4. Now x plus delta x to the power 1 by 4 is 15 to the power 1 by 4 and x to the power 1 by 4 is 16 to the power 1 by 4. So this implies 15 to the power 1 by 4 is equal to delta y plus 16 to the power 1 by 4. 16 can be written as 2 to the power 4 and its power is 1 by 4 so this is equal to delta y plus 2. Now we know that delta y is approximately equal to dy and dy is equal to dy by dx into delta x so delta y is equal to dy by dx into delta x. Now y is x to the power 1 by 4 so dy by dx will be 1 by 4 into x to the power 1 by 4 minus 1 that is minus 3 by 4 into delta x and delta x here is minus 1 so this is equal to 1 by 4 into x to the power 3 by 4 into minus 1. Now again substitute the value of x and x is 16 which can be written as 2 to the power 4 and its power is 3 by 4 so now we have 2 to the power 3 into 4 that is 8 into 4 that is 32 into minus 1 so this is equal to minus 1 upon 32 so this is delta y now 15 to the power 1 by 4 is delta y plus 2 so 15 to the power 1 by 4 is equal to minus 1 upon 32 plus 2 that is this is equal to 63 upon 32 that is equal to 1.96875 hence the answer is 1.96875 so this completes the question and the session bye for now take care have a good day