If `y={x+sqrt(x^2+a^2)}^n` , then prove that `(dy)/(dx)=(n y)/(sqrt((x^2+a^2)`

Explanation

We have,



`y= {[x+sqrt{x^{2}+a^{2}}]^{n} }`


`Rightarrow frac{{dy}}{{dx}}=frac{d}{{dx}}[{x+sqrt{x^{2}+a^{2}}}^{n}] `


`= n{x+sqrt{x^{2}+a^{2}}}^{n}-1 frac{d}{{dx}}{x+sqrt{.x^{2}+a^{2}}}.`


`{[text { By chain rule }] } `


`= n{x+sqrt{x^{2}+a^{2}}}^{n-1} cdot{frac{d}{{dx}}(x)+frac{d}{{dx}} sqrt{x^{2}+a^{2}}}`


`= n{x+sqrt{x^{2}+a^{2}}}^{n-1} cdot{1+frac{1}{2}(x^{2}+a^{2})^{frac{-1}{2}} cdot frac{d}{{dx}}(x^{2}+a^{2})} `


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