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यदि u = (8x2 + y2) (log x - log y), तब  बराबर है

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  1. u/2
  2. 2u
  3. u
  4. -2u

Answer (Detailed Solution Below)

Option 2 : 2u
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अवधारणा:

एक से अधिक स्वतंत्र चर के एक फलन को प्रत्येक स्वतंत्र चर के लिए अलग-अलग आंशिक रूप से अवकलन किया जा सकता है।

उदाहरण के लिए, यदि f(x, y) = 2x2 + 3y2

x के सापेक्ष f(x, y) का आंशिक अवकल fx =  =  = 4x है (अवकलन होने पर y को एक स्थिरांक के रूप में लें)

y के सापेक्ष f(x, y) का आंशिक अंतर fy =  = = 6y है (अवकलन करते समय x को एक अचर के रूप में लें)

गणना:

दिया गया है:

समीकरण u = (8x2 + y2)(log x - log y) है। 

अभीष्ट आंशिक अवकलों की गणना नीचे की गई है:

ux =  16x(log x - log y) + (8x2 + y2)() = 16xlog x - 16xlog y + 8x + 

uy = 2y(log x - log y) + (8x2 + y2)(-) = 2ylog x - 2ylog y - - y

  • परिणाम की गणना नीचे की गई है:

x+y

= x(16xlog x - 16xlog y + 8x + ) + y(2ylog x - 2ylog y - - y)

= 16x2log x - 16x2log y + 8x2 + y2 + 2y2log x - 2y2log y - 8x2 - y2

= 16x2log x + 2y2log x - 16x2log y - 2y2log y

= 2log x(8x2 + y2) - 2log y(8x2 + y2)

= 2(8x2 + y2)(log x - log y)

= 2u

अतः, सही उत्तर विकल्प 2 है। 

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