Math 130: Written Homework 4 Solution due Friday, March 3 Name: Problem Statement (Derivative Rules) Derivative rules are extremely useful and make taking the derivative a much faster process than using the limit definition. However, it is from the limit definition that derivative rules come from. Below, you will be proving two derivative rules using the limit definition of the derivative: 1. Prove d = nx"-using the limit definition of the derivative. dx Hint: x7 - an = (x - a)(x-1+ x-2a+ 27-3a2+ ...+ x2an-3 + xan-2+ an-1) 2. Prove dx (f(x)g(x)) = f'(x)g(x) + f(x)g'(x) using the limit definition of the derivative. Hint: Rewrite the numerator from the limit definition of the derivative as: f ( x + h)g(x + h) - f(x + h)g(x) + f(x + h)g(x) - f(x)g(x)