Derivatives:
\(f^0(c) = \frac{1}{(1-c)}\) \(f'(c) = \frac{1}{(1-c)^2}\) \(f''(c) = \frac{2}{(1-c)^3}\) \(f'''(c) = \frac{6}{(1-c)^4}\)
The Taylor series would be defined as \(f(x) = \frac{1}{(1-c)0!}(x-c)^0 + \frac{1}{(1-c)^2*1!}(x-c)^1 + \frac{2}{(-c)^3*2!}(x-c)^2 + \frac{6}{(1-c)^4*3!}(x-c)^3...+\sum_{n=0}^\infty \frac{1}{(1-c)^(n+1)}(x-c)^n\)
Derivatives: \(f^0(c) = e^c\) \(f'(c) = e^c\) \(f''(c) = e^c\)
The Taylor series would be: \(f(x) = \frac{e^c}{0!}(x-c)^0 + \frac{e^c}{1!}(x-c)^1 + \frac{e^c}{2!}(x-c)^2...+\sum_{n=0}^\infty \frac{(x-c)^n}{n!}\)
Derivatives: \(f^0(c) = ln(1+c)\) \(f'(c) = \frac{1}{1+c}\) \(f''(c) = \frac{1}{(1+c)^2}\) \(f'''(c) = \frac{2}{(1+c)^3}\)
The Taylor series would be: \(f(x) = ln(1+x) + \frac{1}{(1+c)1!} + \frac{1}{(1+c)^2*2!} + \frac{2}{(1+c)^3*3!}+ \sum_{n=1}^\infty \frac{(x-c)^n}{n(1+c)^n}\)
Derivatives: \(f^0(c) = c^(1/2)\) \(f'(c) = \frac{1}{2c^(1/2)}\) \(f''(c) = \frac{1}{4c^(3/2)}\) \(f'''(c) = \frac{3}{8c^(5/2)}\)
The Taylor series would be: \(f(x) = c^(1/2) + \frac{1}{2c^(1/2)*1!}(x-1) + \frac{1}{4c^(3/2)*2!}+...\sum_{n=1}^\infty n^(1/2)(x-c)^n(c^(1/2-n)\)