A) \[\frac{ny}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\]
B) \[-\frac{ny}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\]
C) \[\frac{nx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\]
D) \[-\frac{nx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\]
Correct Answer: A
Solution :
[a] \[\frac{dy}{dx}=\frac{d}{dx}\left[ {{(x+\sqrt{{{x}^{2}}+{{a}^{2}}})}^{n}} \right]\] \[=n{{\left( x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right)}^{n-1}}.\frac{d}{dx}\left( x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right)\] \[=n{{\left( x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right)}^{n-1}}\left( \frac{\sqrt{{{x}^{2}}+{{a}^{2}}}+x}{\sqrt{{{x}^{2}}+{{a}^{2}}}} \right)\] \[=\frac{n{{(x+\sqrt{{{x}^{2}}+{{a}^{2}}})}^{n}}}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\] \[=\frac{ny}{\sqrt{{{n}^{2}}+{{a}^{2}}}}\]
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