A) \[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\]
B) \[\frac{x}{({{a}^{2}}+{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\]
C) \[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}+\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\]
D) \[({{a}^{2}}+{{b}^{2}})\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\]
Correct Answer: A
Solution :
\[f(x)=\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}+\sqrt{{{x}^{2}}+{{b}^{2}}}}\] \[f(x)=\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}+\sqrt{{{x}^{2}}+{{b}^{2}}}}.\frac{\sqrt{{{x}^{2}}+{{a}^{2}}}-\sqrt{{{x}^{2}}+{{b}^{2}}}}{\sqrt{{{x}^{2}}+{{a}^{2}}}-\sqrt{{{x}^{2}}+{{b}^{2}}}}\] \[f(x)=\frac{1}{{{a}^{2}}-{{b}^{2}}}\left[ \sqrt{{{x}^{2}}+{{a}^{2}}}-\sqrt{{{x}^{2}}+{{b}^{2}}} \right]\] \[f'(x)=\frac{1}{{{a}^{2}}-{{b}^{2}}}\left[ \frac{2x}{2\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2x}{2\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] \[f'(x)=\frac{x}{{{a}^{2}}-{{b}^{2}}}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\].
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