A) \[{{2}^{x-y}}\frac{{{2}^{y}}-1}{{{2}^{x}}-1}\]
B) \[{{2}^{x-y}}\frac{{{2}^{y}}-1}{1-{{2}^{x}}}\]
C) \[\frac{{{2}^{x}}+{{2}^{y}}}{{{2}^{x}}-{{2}^{y}}}\]
D) None of these
Correct Answer: B
Solution :
On differentiating \[{{2}^{x}}\log 2+{{2}^{y}}\log 2.\frac{dy}{dx}\] \[={{2}^{x}}{{.2}^{y}}\frac{dy}{dx}.\log 2+{{2}^{y}}{{.2}^{x}}\log 2\] Þ \[{{2}^{x}}+{{2}^{y}}\frac{dy}{dx}={{2}^{x+y}}\frac{dy}{dx}+{{2}^{x+y}}\] Þ \[\frac{dy}{dx}=\frac{{{2}^{x+y}}-{{2}^{x}}}{{{2}^{y}}-{{2}^{x+y}}}={{2}^{x-y}}\frac{{{2}^{y}}-1}{1-{{2}^{x}}}\].
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