If f(x)=xn, then the value of f(1)+(f1(1)/1)+(f2(1)/2 !)+⋅s+(fn(1)/n !), where fr(x) denotes the rth order derivative of f(x) with respect to x, is

Question

If f(x)=xn, then the value of f(1)+(f1(1)/1)+(f2(1)/2 !)+⋅s+(fn(1)/n !), where fr(x) denotes the rth order derivative of f(x) with respect to x, is (A) n (B) 2n (C) 2n-1 (D) None of these

Tardigrade Admin · Accepted Answer

We have f(x)=xn . So, f1(x)=n xn-1 ⇒ f prime(1)=n f2(x)=n(n-1) xn-2 ⇒ f2(1)=n(n-1) f3(x)=n(n-1)(n-2) xn-3 ⇒ f3(1)=n(n-1)(n-2) fn(x)=n(n-1)(n-3) ldots 1 ⇒ fn(1)=n(n-1)(n-3) ldots 1 ⇒ f(1)+(f1(1)/1)+(f2(1)/2 !)+⋅s+(fn(1)/n !) =1+(n/1)+(n(n-1)/2 !)+(n(n-1)(n-2)/3 !)+⋅s+(n(n-1)(n-2) ldots 1/n !) = n C0+ n C1+ n C2+⋅s+ n Cn =2n

Q. If $f (x) = x^{n},$ then the value of $f (1) + \frac{f ^{1} ( 1 )}{1} + \frac{f ^{2} ( 1 )}{2 !} + \dots + \frac{f ^{n} ( 1 )}{n !},$ where $f^{r} (x)$ denotes the rth order derivative of $f (x)$ with respect to $x$ , is

$n$

$2^{n}$

$2^{n - 1}$

None of these

Q. If f(x)=xn, then the value of f(1)+1f1(1)​+2!f2(1)​+⋯+n!fn(1)​, where fr(x) denotes the rth order derivative of f(x) with respect to x, is

n

2n

2n−1

None of these

Q. If $f (x) = x^{n},$ then the value of $f (1) + \frac{f ^{1} ( 1 )}{1} + \frac{f ^{2} ( 1 )}{2 !} + \dots + \frac{f ^{n} ( 1 )}{n !},$ where $f^{r} (x)$ denotes the rth order derivative of $f (x)$ with respect to $x$ , is

$n$

$2^{n}$

$2^{n - 1}$