MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the two rational functions
(i) f1(x)=x−2x2−4 and
(ii) f2(x)=x2+13x2+5.
Both functions are differentiable for all values of x in their respective domains.
(1) Without calculating any derivatives, describe which of the two functions would be quicker to differentiate after a preliminary algebraic manipulation of the numerator and denominator. Clearly explain the feature of each function that supports your choice.
(2) For the function you identified in part (a), carry out the recommended manipulation and then find f′(x) by differentiating the resulting expression.
(3) Differentiate the same function directly from its original, unreduced form and verify that the derivative agrees with your answer from part (b).