If `f(x)=x^n ,f(1)+(f^1(1))/1+(f^2(1))/(2!)+(f^n(1))/(n !),w h e r ef^r(x)` denotes the rth order derivative of `f(x)` with respect to `x ,` is a. `n` b. `2^n` c. `2^(n-1)` d. none of these

Options

A. n-1
B. `2^(n)`
C. `2^(n-1)`
D. none of these

Answer

b

Explanation

We have,
`f(x) = x^(n)`
`rArr f'(x)= (n!)/((n-r)!) x^(n-r) rArr f'(1) = (n!)/((n-r)!)`
Now,
`f(1) + (f ""^(1)(1))/(1) +(f""^(2)(1))/(2!) +...+(f^(n) (1))/(n!),`
`= sum_(r=0)^(n) (f ""^(r)(1))/(r!) = sum_(r=0)^(n) (n!)/((n -r)!r!) = sum_(r=0)^(n) ""^(n)C_(r) = 2^(n)` .