MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
The natural logarithm function g(x)=ln(1+x) is defined for x>−1.
(a) Compute g′(x),g′′(x), and g′′′(x). Write each derivative as a single rational expression.
(b) Based on the work in part (a), conjecture a general formula for g(n)(x) for every positive integer n. Your answer should be written as a simplified rational expression in terms of n and x.
(c) Prove that the formula you stated in part (b) is valid for all positive integers n. Structure your argument clearly.
(d) Using your proven formula, determine the exact value of g(10)(0.5). Present your result as a simplified rational number.