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Derivative of $$$- \frac{x}{y}$$$ with respect to $$$x$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{dx} \left(- \frac{x}{y}\right)$$$.
Solution
Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = - \frac{1}{y}$$$ and $$$f{\left(x \right)} = x$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \frac{x}{y}\right)\right)} = {\color{red}\left(- \frac{1}{y} \frac{d}{dx} \left(x\right)\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- \frac{{\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{y} = - \frac{{\color{red}\left(1\right)}}{y}$$Thus, $$$\frac{d}{dx} \left(- \frac{x}{y}\right) = - \frac{1}{y}$$$.
Answer
$$$\frac{d}{dx} \left(- \frac{x}{y}\right) = - \frac{1}{y}$$$A