GATE 2018 EC – Question 62
Let $r=x^2+y-z$ and $z^3-xy+yz+y^3=1$. Assume that $x$ and $y$ are independent variables. At $(x,y,z)=(2,-1,1)$, the value (correct to two decimal places) of $\dfrac{\partial r}{\partial x}$ is ________.
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Correct answer: 4.48 to 4.52
Explanation
Differentiating the constraint with respect to $x$: $3z^2z_x-y+yz_x=0$, so $z_x=\frac{y}{3z^2+y}=\frac{-1}{3-1}=-\frac12$. Then $\frac{\partial r}{\partial x}=2x-z_x=4+\frac12=4.5$.