Question

If $u = e^x \sin y$ then first partial derivative of $u$ with respect to $y$ is

• A. $e^x \sin y$
• B. $e^x \cos y$
• C. $-e^x \cos y$
• D. $0$

Hint:

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.

Answer 1

The Correct Option is B

In partial differentiation, when we differentiate with respect to one variable, all other independent variables are treated as constants. 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 $y$.