Derivative of $$$\frac{y}{x}$$$ with respect to $$$y$$$
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Find $$$\frac{d}{dy} \left(\frac{y}{x}\right)$$$.
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 $$$c = \frac{1}{x}$$$ and $$$f{\left(y \right)} = y$$$:
$${\color{red}\left(\frac{d}{dy} \left(\frac{y}{x}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dy} \left(y\right)}{x}\right)}$$Apply the power rule $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dy} \left(y\right) = 1$$$:
$$\frac{{\color{red}\left(\frac{d}{dy} \left(y\right)\right)}}{x} = \frac{{\color{red}\left(1\right)}}{x}$$Thus, $$$\frac{d}{dy} \left(\frac{y}{x}\right) = \frac{1}{x}$$$.
Answer
$$$\frac{d}{dy} \left(\frac{y}{x}\right) = \frac{1}{x}$$$A