Question textLet [math: f(x,y)=ycos⁡(xy)]f(x,y) = {y\,\cos \left( x\,y \right)}. The partial derivative [math: ∂f∂x]\frac{\partial f}{\partial {x}} is[input] Your last answer was interpreted as follows: rickli12128@outlook.comThis answer is invalid. Expected "!!", "!", "#", "#pm#", "%and", "%or", "(", "*", "**", "+", "+-", ",", "-", ".", "/", ":", "::", "::=", ":=", "=", "@@IS@@", "@@Is@@", "[", "^", "^^", "and", "implies", "nand", "nor", "nounand", "nounor", "or", "xnor", "xor", "~", [;$], end of input or whitespace but "@" found.Your answer should contain the variables [math: x] and [math: y].Check Question 66

Question

Question textLet [math: f(x,y)=ycos⁡(xy)]f(x,y) = {y\,\cos \left( x\,y \right)}. The partial derivative [math: ∂f∂x]\frac{\partial f}{\partial {x}} is[input] Your last answer was interpreted as follows: rickli12128@outlook.comThis answer is invalid. Expected "!!", "!", "#", "#pm#", "%and", "%or", "(", "*", "**", "+", "+-", ",", "-", ".", "/", ":", "::", "::=", ":=", "<", "<=", "=", ">", ">=", "@@IS@@", "@@Is@@", "[", "^", "^^", "and", "implies", "nand", "nor", "nounand", "nounor", "or", "xnor", "xor", "~", [;$], end of input or whitespace but "@" found.Your answer should contain the variables [math: x] and [math: y].Check Question 66

BlackTom AI · Accepted Answer

Let me restate the problem in my own words to ensure clarity: Given f(x, y) = y cos(xy), we want to find the partial derivative of f with respect to x, that is ∂f/∂x.

First, observe that y is treated as a constant with respect to x when taking the partial derivative, since partial derivatives hold other variables constant.

Now apply the chain rule to the inner function cos(xy). The derivative of cos(u) with respect to u is -sin(u), and here u = xy, which itself depends on x via the factor y. Therefore:

∂/∂x [cos(xy)] = -sin(xy) * ∂/∂x [xy].

Compute ∂/∂x [xy]. Since y is constant with respect to x, the derivative is y.

Putting it together: ∂/∂x [cos(xy)] = -sin(xy) * y = -y sin(xy).

Finally, multiply by the outer factor y (which is treated as a constant with respect to x):

∂f/∂x = y * ( -y sin(xy) ) = - y^2 sin(xy).

Key checks and potential misunderstandings:
- If you forget the inner derivative ∂/∂x (xy) = y, you would not get the correct factor of y, leading to an incorrect result.
- If you mistakenly treat y as a function of x, you would incorrectly add extra terms; however, in partial derivatives, y is held constant.
- Some students might drop the negative sign from the derivative of cosine; remember d/dx cos(u) = -sin(u) · du/dx.

Since the provided input does not include multiple answer options to analyze (the answer_options field is empty), there is no list of choices to evaluate for correctness or incorrectness. The result for ∂f/∂x is simply -y^2 sin(xy). If you need me to format this as a step-by-step comparison against a given set of options, please provide the options and I will map this derivation to each choice accordingly.

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Question textLet [math: f(x,y)=ysin⁡(xy)]f(x,y) = {y\,\sin \left( x\,y \right)}. The partial derivative [math: ∂f∂y]\frac{\partial f}{\partial {y}} is[input] Your answer should contain the variables [math: x] and [math: y].Check Question 66

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