A) \[\frac{-1}{{{x}^{2}}+{{b}^{2}}}+\frac{1}{{{x}^{2}}+{{a}^{2}}}\]
B) \[\frac{1}{{{x}^{2}}+{{b}^{2}}}+\frac{1}{{{x}^{2}}+{{a}^{2}}}\]
C) \[\frac{1}{{{x}^{2}}+{{b}^{2}}}-\frac{1}{{{x}^{2}}+{{a}^{2}}}\]
D) none of these
Correct Answer: B
Solution :
Let \[y=\frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)+\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)\] \[\therefore \frac{dy}{dx}=\frac{1}{b}\frac{1}{1+\frac{{{x}^{2}}}{{{b}^{2}}}}\times \frac{1}{b}+\frac{1}{a}\times \frac{1}{1+\frac{{{x}^{2}}}{{{a}^{2}}}}\times \frac{1}{a}\] \[=\frac{1}{{{b}^{2}}+{{x}^{2}}}+\frac{1}{{{a}^{2}}+{{x}^{2}}}\]
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