MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the function g(x)=x5. Provide a complete verification that the derivative of g at an arbitrary point x satisfies g′(x)=5x4. Your response must rely solely on algebraic manipulation and limit evaluation.
(a) Write the difference quotient expression that defines g′(x).
(b) Expand (x+h)5 using an appropriate algebraic theorem, and organize the result so that all terms are visible.
(c) Simplify the numerator of the difference quotient, factor out all common factors of h, and write the resulting fraction.
(d) Evaluate the limit as h→0 and state the final formula for g′(x), explicitly explaining why every term except one vanishes in the limit.