KCET2023MathematicsDifferentiation
If f(x)=1+n x+ n(n-1) 2 x^2+ n(n-1)(n-2) 6 x^3+ +x^n , then f^n(1) is equal to
Options
- An(n-1) 2^ n-2
- Bn(n-1) 2^n
- C2^ n-1
- D(n-1) 2^ n-1
Correct answer
A. n(n-1) 2^ n-2
Step-by-step solution
Given, aligned f(x)=1+n x+ n(n-1) 2 x^2+ n(n-1)(n-2) 6 x^3 & & + +x^n . aligned The given equation can be written as f(x)=(1+x)^n (1+x)^n=1+n x+ n(n-1) 2 ! x^2+ n(n-1)(n-2) 3 ! x^3+ x^n Now, f^ (x)=n(1+x)^ n-1 and, f^ (x)=n(n-1)(1+x)^ n-2 put x=1 f^ (1)=n(n-1) 2^ n-2