Differentiate following functions.

Solution

At this point there in fact isn't a lot of cause to use the product rule. 

We will utilize the product rule.  As we add up more functions to our repertoire and as the functions become more complexes the product rule will become more useful and in several cases required.

Note as well that we took the derivative of this function in the previous section and didn't use the product rule at that point.  However, we have to get the same result here as we did then.

By converting the radical to a fractional exponent as always, we get.

                                                y = x ^2/3 (2 x - x² )

Now let's take the derivative.  Hence we take the derivative of the first function times the second then add up on to that the first function times the derivative of the second function.

                                         y′ = (2/3) x ^-1/3 (2 x - x² ) + x ^2/3 ( 2 - 2 x )

                          y′ =(4/3)x^(2/3)-(2/3) x^(5/3) +2x ^(2/3) -2x ^(5/3) =(10/3) x^(2/3) -(8/3)x^(5/3)