A) \[\frac{5}{1+25{{x}^{2}}}\]
B) \[\frac{1}{1+25{{x}^{2}}}\]
C) \[0\]
D) \[\frac{5}{1-25{{x}^{2}}}\]
E) None of these
Correct Answer: A
Solution :
Given, \[y={{\tan }^{-1}}\left( \frac{4x}{1+5{{x}^{2}}} \right)+{{\tan }^{-1}}\left( \frac{2+3x}{3-2x} \right)\] Or \[y={{\tan }^{-1}}\left( \frac{5x-x}{1+5{{x}^{2}}} \right)+{{\tan }^{-1}}\left( \frac{\frac{2}{3}+x}{1-\frac{2}{3}x} \right)\] \[\Rightarrow \]\[y={{\tan }^{-1}}(5x)+{{\tan }^{-1}}(x)\] \[+{{\tan }^{-1}}\left( \frac{2}{3} \right)+{{\tan }^{-1}}(x)\] \[\Rightarrow \]\[y={{\tan }^{-1}}(5x)+{{\tan }^{-1}}\left( \frac{2}{3} \right)\] On differentiating w.r.t.\[x,\]we get \[\frac{dy}{dx}=\frac{5}{1+{{(5x)}^{2}}}=\frac{5}{1+25{{x}^{2}}}\]
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