 Hello and welcome to the session. In this session we are going to discuss differential coefficient of x raised to power n and a x plus b whole raised to power n where x is greater than 0 and x belonging to r that is the set of real numbers n is an integer or a rational number. First, let y be equal to x raised to power n. Mark this equation as 1. Let delta x be increment in x delta y because coming increment in y so that y plus delta y is equal to x plus delta x by raised to power n. Mark this equation as 2. Subtract equation 1 from equation 2 we get delta y is equal to x plus delta x by raised to power n minus x raised to power n or delta y can be written as x raised to power n into 1 plus delta x by x by raised to power n minus x raised to power n here delta x is small therefore delta x by x may be considered less than 1 numerically therefore applying binomial theorem for any index we get delta y is equal to x raised to power n into 1 plus n into delta x by x plus n into n minus 1 upon 1 into 2 by multiplied by delta x by x b whole square plus terms containing higher power of delta x minus 1 therefore we have delta y is equal to x raised to power n into taking delta x common we have delta x into n upon x plus n into n minus 1 upon 1 into 2 by multiplied by delta x by x square plus terms containing delta x whole square and higher power of delta x. Now dividing both the sides by delta x delta y by delta x is equal to x raised to power n into n upon x plus n into n minus 1 by 1 into 2 by multiplied by delta x by x square plus terms containing delta x square and higher power of delta x. Now taking limits as delta x tends to 0 we have limit delta y by delta x as delta x tends to 0 is equal to limit x raised to power n into n by x plus n into n minus 1 upon 1 into 2 into delta x by x square plus terms containing delta x square and higher power of delta x as delta x tends to 0. Therefore we have limit delta y by delta x as delta x tends to 0 can be written as dy by dx is equal to now putting the value of delta x as 0 in the right hand side of the equation we get x raised to power n into n upon x plus 0 which implies that dy by dx is equal to n into x raised to power n minus 1. Therefore we have dy by dx of y that is x raised to power n is given by n into x raised to power n minus 1 where x belongs to the set of real numbers are and x is greater than 0 term integer as a rational number. For example we have dy by dx of x raised to power 8 which can be written as 8 into x raised to power 8 minus 1 that is 8 into x raised to power 7 and the value of dy by dx of x raised to power 5 by 2 is equal to 5 by 2 into x raised to the power 5 by 2 minus 1 that is equal to 5 by 2 into x raised to power 3 by 2. Now again let y be equal to a into x plus b raised to power n and mantis equation as 1. Let delta x be the increment in x delta y the corresponding increment in y so that y plus delta y is equal to a into x plus delta x plus b raised to power n and mantis equation as 2. Subtracting equation 1 from equation 2 we have delta y is equal to a into x plus delta x plus b raised to power n minus of a into x plus b raised to power n delta y is equal to a into x plus a into delta x plus b raised to power n minus of a into x plus b raised to power n. We can also write delta y as a x plus b plus a delta x raised to power n minus a x plus b raised to power n. And we can write it as delta y is equal to now taking a into x plus b raised to power n common from both the terms we get a into x plus b raised to power n into 1 plus a into delta x upon a into x plus b raised to power n minus 1 delta y is equal to a into x plus b raised to power n into 1 plus a into delta x upon a into x plus b raised to power n can be written using binomial theorem as 1 plus n into a into delta x upon a into x plus b plus n into n minus 1 upon 1 into 2 into a delta x upon a into x plus b raised to the power 2 plus terms containing higher powers of delta x minus 1 by delta y is equal to a into x plus b raised to the power n into n into a into delta x upon a into x plus b plus n into n minus 1 upon 1 into 2 multiplied by a into delta x upon a x plus b the whole square plus terms containing higher powers of delta s. Now delta y is equal to a into x plus b raised to power n and taking delta x common from all the terms we have delta x into n into a upon a into x plus b plus n into n minus 1 upon 1 into 2 multiplied by a square into delta x upon a into x plus b the whole square plus terms containing higher powers of delta x. Now dividing both the sides by delta x we get delta y by delta x is equal to a into x plus b raised to power n into n into a upon a into x plus b plus n into n minus 1 upon 1 into 2 into a square into delta x upon a into a into x plus b the whole square plus terms containing higher powers of delta x taking minutes as delta x tends to 0 limit delta y by delta x as delta x tends to 0 is equal to a limit a into x plus b for delta power n into n into a upon a into x plus b plus n into n minus 1 upon 1 into 2 into a square into delta x upon a into x plus b the whole square plus terms containing higher powers of delta x as delta x tends to 0. Now limit delta y upon delta x as delta x tends to 0 can be written as dy by dx and is equal to now putting the value of delta x as 0 in the right hand side of the equation. Now we get a x plus b raised to power n into n into a upon a into x plus b plus 0 therefore dy by dx is equal to a into x plus b raised to power n into n into a upon a into x plus b which can be written as dy by dx is equal to n into a into a into x plus b raised to power n minus 1 therefore we have dy by dx of y that is a into x plus b raised to power n is equal to n into a into a into x plus b raised to power n minus 1 and we can also write it as n into a into x plus b raised to power n minus 1 into a that is if we have dy by dx of a linear function of x raised to power n then it's equal to n into linear function raised to power n minus 1 into coefficient of x for example if we have dy dx of 3x plus 5 raised to power 7 then this is equal to 7 into 3x plus 5 raised to power 7 minus 1 into coefficient of x that is 3 therefore we have 7 into 3 by 21 into 3x plus 5 raised to the power 6 hence the value of dy dx of 3x plus 5 raised to the power 7 is equal to 21 into 3x plus 5 raised to the power 6 this completes our session hope you enjoyed this session