Question

The first-order partial derivative of the function \( \log(\sin x + \cos y) \) with respect to the variables \( x \) and \( y \), respectively, are given by:

• A. \( \frac{\cos x}{\sin x + \cos y}, \frac{-\sin y}{\sin x + \cos y} \)
• B. \( \frac{\sin x}{\sin x + \cos y}, \frac{\cos y}{\sin x + \cos y} \)
• C. \( \frac{-\cos x}{\sin x + \cos y}, \frac{-\sin y}{\sin x + \cos y} \)
• D. \( \frac{\cos x}{\sin x + \cos y}, \frac{\sin y}{\sin x + \cos y} \)

Hint:

Apply the chain rule carefully for logarithmic functions involving multiple variables.

Answer 1

The Correct Option is A

Using partial differentiation:

\( \frac{\partial}{\partial x} \log(\sin x + \cos y) = \frac{\cos x}{\sin x + \cos y}, \quad \frac{\partial}{\partial y} \log(\sin x + \cos y) = \frac{-\sin y}{\sin x + \cos y}. \)