Derivative of $$$- c + c_{max}$$$ with respect to $$$c$$$
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Your Input
Find $$$\frac{d}{dc} \left(- c + c_{max}\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dc} \left(- c + c_{max}\right)\right)} = {\color{red}\left(- \frac{d}{dc} \left(c\right) + \frac{dc_{max}}{dc}\right)}$$Apply the power rule $$$\frac{d}{dc} \left(c^{n}\right) = n c^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dc} \left(c\right) = 1$$$:
$$- {\color{red}\left(\frac{d}{dc} \left(c\right)\right)} + \frac{dc_{max}}{dc} = - {\color{red}\left(1\right)} + \frac{dc_{max}}{dc}$$The derivative of a constant is $$$0$$$:
$${\color{red}\left(\frac{dc_{max}}{dc}\right)} - 1 = {\color{red}\left(0\right)} - 1$$Thus, $$$\frac{d}{dc} \left(- c + c_{max}\right) = -1$$$.
Answer
$$$\frac{d}{dc} \left(- c + c_{max}\right) = -1$$$A
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