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GATE 2014 EC (EC3) – Question 15

Engineering Mathematics · Calculus · 1 mark · Multiple choice

If $z=xy\ln(xy)$, which derivative identity holds?

  1. $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0$
  2. $y\frac{\partial z}{\partial x}=x\frac{\partial z}{\partial y}$
  3. $x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}$
  4. $y\frac{\partial z}{\partial x}+x\frac{\partial z}{\partial y}=0$

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Correct answer: (C) $x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}$

Explanation

On the real domain xy>0, differentiate to obtain zx=y[ln(xy)+1] and zy=x[ln(xy)+1]. Multiplying respectively by x and y gives equal expressions xy[ln(xy)+1]. Therefore **x∂z/∂x=y∂z/∂y,C**.