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