MCQ
If  $\ln \left( {\left( {e - 1} \right){e^{xy}} + {x^2}} \right) = {x^2} + {y^2}$ , then ${\left. {\frac{{dy}}{{dx}}} \right|_{\left( {1,0} \right)}}$     is
  • A
    $0$
  • B
    $1$
  • $2$
  • D
    $4$

Answer

Correct option: C.
$2$
c
Differentiate both sides wrt $'x',$

$(e - 1){e^{xy}}\left( {\frac{{xdy}}{{dx}} + y} \right) + 2x = {e^{{x^2} + {y^2}}}\left( {2x + 2y\frac{{dy}}{{dx}}} \right)$

$\left.(e-1)\left(\frac{d y}{d x}\right)\right|_{(1,0)}+2=e(2)$

$\left.\frac{d y}{d x}\right|_{(1,0)}=2$

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