Derivative of $$$e^{x} \cos{\left(y \right)}$$$ with respect to $$$y$$$
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Your Input
Find $$$\frac{d}{dy} \left(e^{x} \cos{\left(y \right)}\right)$$$.
Solution
Apply the constant multiple rule $$$\frac{d}{dy} \left(c f{\left(y \right)}\right) = c \frac{d}{dy} \left(f{\left(y \right)}\right)$$$ with $$$c = e^{x}$$$ and $$$f{\left(y \right)} = \cos{\left(y \right)}$$$:
$${\color{red}\left(\frac{d}{dy} \left(e^{x} \cos{\left(y \right)}\right)\right)} = {\color{red}\left(e^{x} \frac{d}{dy} \left(\cos{\left(y \right)}\right)\right)}$$The derivative of the cosine is $$$\frac{d}{dy} \left(\cos{\left(y \right)}\right) = - \sin{\left(y \right)}$$$:
$$e^{x} {\color{red}\left(\frac{d}{dy} \left(\cos{\left(y \right)}\right)\right)} = e^{x} {\color{red}\left(- \sin{\left(y \right)}\right)}$$Thus, $$$\frac{d}{dy} \left(e^{x} \cos{\left(y \right)}\right) = - e^{x} \sin{\left(y \right)}$$$.
Answer
$$$\frac{d}{dy} \left(e^{x} \cos{\left(y \right)}\right) = - e^{x} \sin{\left(y \right)}$$$A