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Chapter 13: Problem 9

Find the first partial derivatives of the following functions. $$f(x, y)=3 x^{2} y+2$$

Short Answer

Expert verified
Question: Find the first partial derivatives of the function $$f(x, y) = 3x^2y + 2$$. Answer: The first partial derivatives are $$\frac{\partial f}{\partial x} = 6xy$$ and $$\frac{\partial f}{\partial y} = 3x^2$$.

Step by step solution

01

Differentiate with respect to x

Keep y constant and find the derivative of the function with respect to x: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(3x^2y + 2)$$
02

Compute the x partial derivative

Applying the power and constant rule for derivatives, we have: $$\frac{\partial f}{\partial x} = 6xy$$
03

Differentiate with respect to y

Keep x constant and find the derivative of the function with respect to y: $$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(3x^2y + 2)$$
04

Compute the y partial derivative

Applying the power and constant rule for derivatives, we have: $$\frac{\partial f}{\partial y} = 3x^2$$
05

Write down the first partial derivatives

The first partial derivatives of the function $$f(x, y) = 3x^2y + 2$$ are: $$\frac{\partial f}{\partial x} = 6xy$$ $$\frac{\partial f}{\partial y} = 3x^2$$

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