If $u = e^x \sin y$ then first partial derivative of $u$ with respect to $y$ is
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Always double-check which variable you are differentiating against. If this question asked for $\frac{\partial u}{\partial x}$, the answer would be $e^x \sin y$ because $\sin y$ would be treated as the constant.
1. Identify the variables: The function is $u(x, y) = e^x \sin y$.
We need to find $\frac{\partial u}{\partial y}$.
2. Apply the Partial Derivative: When differentiating with respect to $y$, the term $e^x$ is treated as a constant factor.
$$\frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(e^x \sin y)$$
$$\frac{\partial u}{\partial y} = e^x \cdot \frac{\partial}{\partial y}(\sin y)$$
The derivative of $\sin y$ with respect to $y$ is $\cos y$.
$$\frac{\partial u}{\partial y} = e^x \cos y$$
This is the first partial derivative of the function with respect to $y$.