Find the derivative of the function f(x) given by

Here,

$f(x)=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)$

$f(1)=(2)(2)(2)(2)=16$

Taking log on both sides we get,

$\log (f(x))=\log (1+x)+\log \left(1+x^{2}\right)+\log \left(1+x^{4}\right)+\log \left(1+x^{8}\right)$

Differentiating it with respect to $x$ we get,

$\frac{1}{f(x)} \frac{d(f(x))}{d x}=\frac{1}{x+1}+\frac{1}{1+x^{2}} 2 x+\frac{1}{1+x^{4}} 4 x^{3}+\frac{1}{1+x^{8}} 8 x^{7}$

$f^{\prime}(x)=f(x)\left[\frac{1}{x+1}+\frac{1}{1+x^{2}} 2 x+\frac{1}{1+x^{4}} 4 x^{3}+\frac{1}{1+x^{8}} 8 x^{7}\right]$

$f^{\prime}(1)=f(1)\left[\frac{1}{2}+\frac{2}{1+1}+\frac{4}{1+1}+\frac{8}{1+1}\right]$

$f^{\prime}(1)=16\left[7+\frac{1}{2}\right]$

$f^{\prime}(1)=16 \times \frac{15}{2}$

$F^{\prime}(1)=120$