If `y=(1+x)(1+x^2)(1+x^4)(1+x^8)dot(1+x^2^n)` , find `(dy)/(dx)` .

Explanation

We have `y=(1+x)(1+x^{2})(1+x^{4}) ldots ldots(1+x^{4})`

take natural logarithm both sides


ln `y=ln (1+x)+ln (1+x^{2})+ln (1+x^{4})+ldots ldots ldots ldots ldots .+ln (1+x^{2 n})`


Now differentiate both sides w.r.t. `x`


`Rightarrow frac{1}{y} frac{d y}{d x}=frac{1}{1+x}+frac{2 x}{1+x^{2}}+frac{4 x^{3}}{1+x^{4}}+ldots . .+frac{2 n x^{2 n-1}}{1+x^{2 n}}`
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