$$
f(x, y, z)=\left(x y, \ln (x z), y^{2} z^{2}\right)
$$

What is $\frac{\partial f}{\partial y} ?$
Choose 1 answer:
(A) $\left(x, 0,2 y z^{2}\right)$
(B) $\left(x+y, 1,2 y z^{2}+2 z y^{2}\right)$
(C) $\left(y, \frac{1}{x}, 0\right)$
(D) $\left(0, \frac{1}{z}, 2 z y^{2}\right)$

Question

$$
f(x, y, z)=\left(x y, \ln (x z), y^{2} z^{2}\right)
$$

What is $\frac{\partial f}{\partial y} ?$
Choose 1 answer:
(A) $\left(x, 0,2 y z^{2}\right)$
(B) $\left(x+y, 1,2 y z^{2}+2 z y^{2}\right)$
(C) $\left(y, \frac{1}{x}, 0\right)$
(D) $\left(0, \frac{1}{z}, 2 z y^{2}\right)$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the partial derivative of the first component $xy$ with respect to $y$, we treat $x$ as a constant. ‖ ‖ step 2 ⋮ The partial derivative of $xy$ with respect to $y$ is $x$. ‖ ‖ step 3 ⋮ To find the partial derivative of the second component $\ln(xz)$ with respect to $y$, we note that there is no $y$ present in the expression, so the derivative is $0$. ‖ ‖ step 4 ⋮ The partial derivative of $y^2z^2$ with respect to $y$ is obtained by treating $z$ as a constant and differentiating $y^2$ to get $2yz^2$. ‖ ‖ step 5 ⋮ Combining the results from steps 2, 3, and 4, we get the partial derivative of $f$ with respect to $y$ as $(x, 0, 2yz^2)$. ‖ ***A*** ∻Key Concept∻ ⚹Partial Derivatives⚹ ∻Explanation∻ ⚹When taking the partial derivative of a function with respect to a variable, treat all other variables as constants and differentiate normally with respect to the chosen variable.◊What is the definition of the partial derivative of a function with respect to a variable?⍭ Generate me a similar question◊⚹

Sia