Question

If the \[\begin{array}{*{20}{c}}
  {f(x)}& = &{1 + nx + \frac{{n(n - 1)}}{{2!}}{x^2} + \frac{{n(n - 1)(n - 2)}}{{3!}}{x^3} + ...{x^n}}
\end{array}\] , then will be,
A) \[n(n - 1){2^{n - 1}}\]
B) \[(n - 1){2^{n - 1}}\]
C) \[n(n - 1){2^{n - 2}}\]
D) \[n(n - 1){2^n}\]

Answer 1

Hint:In this question, the given function f(x) is the expansion of \[{(1 + x)^n}\]. So, replace the function by\[{(1 + x)^n}\]. And after that find the first derivative of the function and the then second derivative of the function. And then put the \[\begin{array}{*{20}{c}}
  x& = &1
\end{array}\].

Formula used:
\[\begin{array}{*{20}{c}}
  {\dfrac{{d\left( {{x^n}} \right)}}{{dx}}}& = &{n{x^{n - 1}}}
\end{array}\]

Complete step by step Solution:
We have given that the function
\begin{array}{*{20}{c}}
  { \Rightarrow f(x)}& = &{1 + nx + \frac{{n(n - 1)}}{{2!}}{x^2} + \frac{{n(n - 1)(n - 2)}}{{3!}}{x^3} + ...{x^n}}
\end{array}
We know that
\begin{array}{*{20}{c}}
  { \Rightarrow {{(1 + x)}^n}}& = &{1 + nx + \frac{{n(n - 1)}}{{2!}}{x^2} + \frac{{n(n - 1)(n - 2)}}{{3!}}{x^3} + ...{x^n}}
\end{array}
Therefore, from equation (a), we will get.
\begin{array}{*{20}{c}}
  { \Rightarrow f(x)}& = &{{{(1 + x)}^n}}
\end{array}
Now, differentiate the above equation with respect to the x. we will get
\begin{array}{*{20}{c}}
  { \Rightarrow f'(x)}& = &{{{n(1 + x)}^{n-1}}}
\end{array}
And again differentiate the above equation with respect to the x.
\begin{array}{*{20}{c}}
  { \Rightarrow f''(x)}& = &{{{n(n-1)(1 + x)}^{n-2}}}
\end{array}…….. (a).
Here, we have given that the value of x is equal to one, So,
 now put the \[\begin{array}{*{20}{c}}
  x& = &1
\end{array}\]in the above equation. therefore, we will get it.
Now, the final answer is \[n(n - 1){2^{n - 2}}\].

Therefore, the correct option is (C).

Additional Information:Differentiation is the technique to elaborate the rate of change in the function with respect to its variable. In other words, we can say that differentiation is a process to determine the rate of change of the function with respect to its variable.
It shows how many changes are occurring in the function with respect to the variable.

Note:The first point to keep in mind is that the expansion of \[{(1 + x)^n}\] is given in the function. Therefore, replace the function by \[{(1 + x)^n}\]. And then differentiate the function with respect to the x. after getting the first derivative of the function f(x). Find the second derivative of the function $f’(x)$. And then put the value of x. to reach the desired result.