Problem 8
Question
Find the first partial derivatives of the following functions. $$f(x, y)=x^{2} y$$
Step-by-Step Solution
Verified Answer
Question: Find the first partial derivatives of the function $$f(x, y) = x^2y$$.
Answer: The first partial derivatives of the function $$f(x, y) = x^2y$$ are:
$$\frac{\partial f}{\partial x} = 2xy$$
$$\frac{\partial f}{\partial y} = x^2$$
1Step 1: Write down the function
We have the function $$f(x, y) = x^2y$$.
2Step 2: Find the partial derivative with respect to x
To find the partial derivative with respect to x, we treat y as a constant and differentiate with respect to x:
$$\frac{\partial f}{\partial x} = \frac{d(x^2y)}{dx} = 2xy$$
3Step 3: Find the partial derivative with respect to y
To find the partial derivative with respect to y, we treat x as a constant and differentiate with respect to y:
$$\frac{\partial f}{\partial y} = \frac{d(x^2y)}{dy} = x^2$$
4Step 4: Write out the first partial derivatives
The first partial derivatives of the function $$f(x, y) = x^2y$$ are:
$$\frac{\partial f}{\partial x} = 2xy$$
$$\frac{\partial f}{\partial y} = x^2$$
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