Suppose g(x)=tanx. Numerical evidence suggests that many derivatives of g evaluated at x=0 are zero.
The tasks below invite you to explore and explain this phenomenon.
a. Calculate g′(0), g′′(0), g′′′(0), and g(4)(0) exactly. Show all intermediate steps in your work.
b. Describe the pattern you observe regarding which of these derivatives are zero and which are non‐zero. State the pattern clearly in words.
c. Provide a conceptual explanation, based on symmetry considerations of tanx and its successive derivatives near x=0, that justifies why all even‐order derivatives of g vanish at x=0 while all odd‐order derivatives are non‐zero.