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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $u = {\tan ^{ - 1}}\left( {{{{x^3} + {y^3}} \over {x - y}}} \right)$, then $x{{\partial u} \over {\partial x}} + y{{\partial u} \over {\partial y}} = $
  • ✓
    $\sin 2u$
  • B
    $\cos 2u$
  • C
    $\tan 2u$
  • D
    $\sec 2u$
✓

Answer

Correct option: A.
$\sin 2u$
a
(a) $\tan u$ is homogeneous in  $x, y $ of degree $ 2.$

$\therefore $ $x\frac{\partial }{{\partial x}}(\tan u) + y\frac{\partial }{{\partial y}}(\tan u) = 2(\tan u)$

$\therefore $ $x{\sec ^2}u\frac{{\partial u}}{{\partial x}} + y{\sec ^2}u\frac{{\partial u}}{{\partial y}} = 2\tan u$

==> $x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}} = 2\frac{{\tan u}}{{{{\sec }^2}u}}$ = $2\sin u\cos u = \sin 2u$.

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