Question

Differentiate
${(x+\frac{1}{x})}^n+n^{1+\frac{1}{x}}$

Answer 1

Consider given the expression,

$\Rightarrow {(x+\frac{1}{x})}^n+n^{1+\frac{1}{x}}$

Differentiate with respect to x,

$\Rightarrow \frac{d}{dx}{(x+\frac{1}{x})}^n+\frac{d}{dx}{n}^{1+\frac{1}{x}}$

$\because \frac{d}{dx}{x}^n=n.x^{n-1}and\frac{d}{dx}{a}^x=a^x{\log }_{e}x$

$\Rightarrow \therefore {(x+\frac{1}{x})}^{n-1}\frac{d}{dx}(x+\frac{1}{x})+n^{1+\frac{1}{x}}.\log {}_{e}n.\frac{d}{dx}(1+\frac{1}{x})$

$\Rightarrow {(x+\frac{1}{x})}^{n-1}(1-\frac{1}{x^2})+n^{1+\frac{1}{x}}.\log {}_{e}n.(0-\frac{1}{x^2})$

$\Rightarrow {(x+\frac{1}{x})}^{n-1}(1-\frac{1}{x^2})-\frac{n^{1+\frac{1}{x}}.\log {}_{e}n}{x^2}$

Hence, this is the answer.