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An air pollution index is $p=x^2+2xy+4xy^2$. At the point $(10,5)$, the partial derivatives $\dfrac{\partial p}{\partial x}$ and $\dfrac{\partial p}{\partial y}$ are:
A420 and 130
B120 and 420
C130 and 400
D130 and 420
Answer & Solution
Correct answer: D. 130 and 420
1. Differentiate with respect to $x$, treating $y$ as constant: $\dfrac{\partial p}{\partial x}=2x+2y+4y^2$.
2. At $(10,5)$: $20+10+4(25)=20+10+100=130$.
3. Differentiate with respect to $y$, treating $x$ as constant: $\dfrac{\partial p}{\partial y}=2x+8xy$.
4. At $(10,5)$: $20+8(10)(5)=20+400=420$.
5. Substituting the point before differentiating, a common slip, leaves an $x$ or $y$ still in the answer.
_Source: NECTA ACSEE 2023 Advanced Mathematics 142/1, Question 10: Differentiation_
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