# Derivatives At 0

Calculus Level 3

Let $$f(x)$$ be continuous and differentiable everywhere, and assume that $$f(x)$$ is analytic. Furthermore assume that for all non-negative integers $$n$$, we have $$f^{(n)}(0)=n$$, where $$f^{(n)}(x)$$ denotes the $$n$$th derivative of $$f$$ at $$x$$. Find $$f(1)$$.

Bonus: As a fun extension, calculate $$f(1)$$ if we say that for non-negative $$n$$, we have $$f^{(n)}(0)=n^k$$ for some defined positive integer $$k$$.