MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Consider the function s(x)=x8. Successive derivatives of s are taken and recorded.
(a) Compute and clearly write the first four derivatives of s, namely s′(x), s′′(x), s′′′(x), and s(4)(x).
(b) Based on your results, describe the pattern you observe in both the numerical coefficient and the exponent after each differentiation step. Provide a verbal explanation connecting your observations to general derivative properties.
(c) Using the pattern you identified, predict s(6)(x) without carrying out a direct differentiation. Explain how your prediction follows logically from the pattern.