Calculate this derivative by definition:

Let      so ze definition of the derivative is    since  and      we can't shove in  yet because we will divide by 0. let us multiply the numerator by its conjugate to rationalize those denominators (no radicals)the conjugate of  is  . let us multiply the numerator and denominator by that now so that we can rationalize and see what happens.    currently we have too many things to deal with. let us focus on the numerator.    since difference of squares factoring is  , multiplying the two conjugates gets us    which simplifies to    if we want to combine these fractions, we need a common denominator.multiply numerator and denominator of by  multiply numerator and denominator of  by      therefore, we plug this back into our difference quotient. then we rewrite the division as multiplication.    we can cancel out the h.    we are now able to take the limit by doing direct substitution.    we can rewrite      thus