 Hello and welcome to the session. Let's work out the following problem. It says using differentials find the approximate value of the following up to three places of decimal. So let us now move on to the solution and let us first define y as a function of x and let us define it as x to the power 1 upon 10 and we choose x so that we can easily find out its 10th root. So we choose x as 1 and delta x as minus of 0.001 since we want to have x plus delta x as 0.999. So x plus delta x is equal to 1 minus 0.001 so this is equal to 0.999. Now we know that delta y is equal to f of x plus delta x minus fx so f of x plus delta x will be x plus delta x to the power 1 by 10 minus fx that is x to the power 1 by 10. Now x plus delta x is 0.999 to the power 1 by 10 minus x to the power 1 by 10 that is 1 to the power 1 by 10 that is 1 only. So this implies 0.999 to the power 1 by 10 is equal to delta y plus 1. Now we know that delta y is approximately equal to dy and dy is equal to dy by dx into delta x so dy or delta y is equal to dy by dx into delta x. Now y is x to the power 1 by 10 so dy by dx will be 1 by 10 into x to the power 1 by 10 minus 1 that is minus 9 by 10 into delta x and the delta x is minus 0.001. Now this is equal to 1 upon 10 into x to the power 9 by 10 into minus 0.001. Now again x is 1 so this becomes 1 by 10 into minus 0.001 and this is equal to minus 0.001. Now this is the value of delta y now 0.999 to the power 1 by 10 is equal to delta y plus 1 now delta y is this that is 0.00001 plus 1 so this is equal to 0.999 hence the value of 0.999 to the power 1 by 10 is 0.999 9 hence the answer is 0.999 9 so this completes the question and the session by for now take care have a good day.