Differentiate the following function `(sqrt(x^2+1)+sqrt(x^2-1))/(sqrt(x^2+1)-sqrt(x^2-1)`

Explanation

begin{aligned}
Let `y=frac{sqrt{x^{2}+1}+sqrt{x^{2}-1}}{sqrt{x^{2}+1}-sqrt{x^{2}-1}} `
`Rightarrow y=frac{sqrt{x^{2}+1}+sqrt{x^{2}-1}}{sqrt{x^{2}+1}-sqrt{x^{2}-1}} times frac{sqrt{x^{2}+1}+sqrt{x^{2}-1}}{sqrt{x^{2}+1}+sqrt{x^{2}-1}} `
`Rightarrow y=frac{(sqrt{x^{2}+1}+sqrt{x^{2}-1})^{2}}{(sqrt{x^{2}+1}-sqrt{x^{2}-1})(sqrt{x^{2}+1}+sqrt{x^{2}-1})}`
`Rightarrow y=frac{(sqrt{x^{2}+1})^{2}+(sqrt{x^{2}-1})^{2}+2 sqrt{(x^{2}+1)(x^{2}-1)}}{(sqrt{x^{2}+1})^{2}-(sqrt{x^{2}-1})^{2}} `
`Rightarrow y=frac{x^{2}+1+x^{2}-1+2 sqrt{(x^{2})^{2}-(1)^{2}}}{(x^{2}+1)-(x^{2}-1)}`
`Rightarrow y=frac{2 x^{2}+2 sqrt{x^{4}-1}}{2}`
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