જો 2^x + 2^y = 2^ x + y , તો dy dx = - Vidyadip

MCQ

જો ${2^x} + {2^y} = {2^{x + y}}$, તો ${{dy} \over {dx}} = $

A
${2^{x - y}}{{{2^y} - 1} \over {{2^x} - 1}}$
✓
${2^{x - y}}{{{2^y} - 1} \over {1 - {2^x}}}$
C
${{{2^x} + {2^y}} \over {{2^x} - {2^y}}}$
D
એકપણ નહીં

✓

Answer

Correct option: B.

${2^{x - y}}{{{2^y} - 1} \over {1 - {2^x}}}$

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$
$\Rightarrow {2^x} + {2^y}\frac{{dy}}{{dx}} = {2^{x + y}}\frac{{dy}}{{dx}} + {2^{x + y}}$
$\Rightarrow \frac{{dy}}{{dx}} = \frac{{{2^{x + y}} - {2^x}}}{{{2^y} - {2^{x + y}}}} = {2^{x - y}}\frac{{{2^y} - 1}}{{1 - {2^x}}}$.

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