 Hi and welcome to the session. I am Deepika here. Let's discuss a question. Differentiate x square minus 5x plus 8 into x cube plus 7x plus 9 in three ways mentioned below. Number one, by using product rule. Number two, by expanding the product to obtain a single polynomial. Number three, by logarithmic differentiation. Do they all give the same answer? So, let us start the solution. That y is equal to our given function which is x square minus 5x plus 8 into x cube plus 7x plus 9. So, start by number one, that is by using product rule. So, we have dy by differentiate both sides with respect to x. We have dy by dx is equal to, now this function that is x square minus 5x plus 8 into derivative of the second function which is 3x square plus 7 plus second function that is x cube plus 7x plus 9 into derivative of the first function which is 2x minus 5. Now, let us solve it we get. So, dy by dx is equal to x square into this is 3x4 plus 7x square minus 15x cube minus 35 plus 24x square plus 56 plus here 4 minus 5x cube plus 14x square minus 35x plus 18x minus 45. So, minus 35x. So, we have dy by dx is equal to 5x raise to power 4 minus 20x cube plus 45x square minus 52x plus 11. So, derivative according to the product rule is 5x4 minus 20x cube plus 45x square minus 52x plus 11. Now, let us start by expanding the product to obtain a single polynomial to by expanding the product to obtain a single polynomial. So, again take that y is equal to our given function y is equal to our given function that is x square minus 5x plus 8 into x cube plus 7x plus 9. So, therefore y is equal to expanded we will get y is equal to x raise to power 5 plus 7x cube plus 9x square minus 5x raise to power 4 minus 35x square minus 45x plus 8x cube plus 56x plus 72. Therefore, y is equal to our x5 minus 5x raise to power 4 plus 15x cube minus 26x square plus 11x plus 72. Therefore, dy by dx is equal to 5x4 minus 20x cube plus 45x square minus 52x plus 11. So, this is the derivative when we expand the product to obtain a single polynomial. So, let us start the third way that is we have to do the above function by logarithmic differentiation. Given y is equal to square minus 5x plus 8 into x cube plus 7x plus 9. Again log on both sides we have log y is equal to log x square minus 5x plus 8 plus log x cube plus 7x plus 9. Now, differentiate both sides with respect to x we get y into derivative of y with respect to x is dy by dx is equal to 1 over x square minus 5x plus 7x plus 9. We get y into derivative of y with respect to x is dy by dx is equal to 1 over x square minus 5x plus 8 into derivative of x square is 2x and this is minus 5 plus 1 over x cube plus 7x plus 9 into derivative of x cube plus 7x that is into 3x square plus 7. This implies dy by dx is equal to y into 2x minus 5 upon x square minus 5x plus 8 plus 3x square plus 7 upon x cube plus 7x plus 9. Substitute value of y here y we get dy by dx is equal to x square minus 5x plus 8 into x cube plus 7x plus 9 into 2x minus 5 upon x square minus 5x plus 8 plus 3x square plus 7 upon x cube plus 7x plus 9. Now, taking LCM and by cancellation we get dy by dx is equal to x cube plus 7x plus 9 into 2x minus 5 plus x square minus 5x plus 8 into 3x square plus 7. So, this is equal to on expanding we get 2x4 minus 5x cube plus 14x square minus 35x plus 18x minus 45 plus plus 3x4 plus 7x square minus 15x plus 9. So, this is equal to 15x cube minus 35x plus 24x square plus 56. So, this is dy by dx. Therefore, dy by dx is equal to 5x4 minus 20x cube plus 45x square plus minus 52x plus 11. Hence, the derivative according to logarithmic differentiation is 5x4 minus 20x cube plus 45x square minus 52x plus 11. Hence, we have found the derivative according to the three mentioned ways and our answer is 5x4 minus 20x cube plus 45x square minus 52x plus 11. So, and they all give the same answer. So, I hope this question is clear to you. Bye and have a nice day.