 Hello and welcome to the session, let's discuss the following question. It says using differentials find approximate value of the following up to 3 places of decimal. So let us now move on to the solution. Let us first define a function y in terms of x and let's take it to be y is equal to cube root of x and here we define or we take x in such a way so that we can easily find out the cube root of x. So we take it to be 0.008 and we take delta x to be 0.001 so that x plus delta x is equal to 0.009. Now we know that delta y is equal to f of x plus delta x minus fx so it is x plus delta x to the power 1 by 3 minus x to the power 1 by 3. Now x plus delta x is 0.009 to the power 1 by 3 and x to the power 1 by 3 is 0.008 to the power 1 by 3. And we have to find the value of 0.009 to the power 1 by 3 so this implies 0.009 to the power 1 by 3 is equal to delta y plus 0.008 can be written as 0.002 to the power 3 and we have its power 1 by 3 so this is equal to delta y plus 0.002. Now we know that delta y is approximately equal to dy so delta y is equal to dy by dx into delta x as dy is equal to dy by dx into delta x. Now x is y is x to the power 1 by 3 so dy by dx will be equal to 1 by 3 into x to the power 1 by 3 minus 1 that is minus 2 by 3 into delta x and delta x is 0.001. Now this is equal to 1 by 3 into x to the power 2 by 3 into 0.001. Now again substitute the value of x we have 3 into 0.008 to the power 2 by 3 into 0.001. Now again this is equal to 1 by 3 into 0.2 to the power 3 and its power 1 by 3 into 0.001 3 gets cancelled with 3 and we have 1 by 3 into 0.2 that is 1 upon 0.6 into 0.001. Here we have power s2 so we will have 0.2 square and 0.2 square is 0.04 and 0.04 into 3 is 0.12 and this is equal to 1 upon 120 and this is equal to 0.008 approximately Now 0.009 to the power 1 by 3 is equal to delta y plus 0.3 and delta y is 0.008 so this is equal to 0.008 plus 0.2 so this is equal to 0.208. Hence the answer is 0.208 so this completes the question and the session. Bye for now take care and have a good day.