I did not include the following in the related post.

Differential of State of Charge Equation

1. Let’s start by simplying the expression. I.E. Multiply through by \(45\). Which gives us:

$$ f(\text{hrs})=(0.8\sin (-\frac{\pi * \text{hrs}}{12})+1) * 45$$ $$=36\sin (-\frac{\pi * \text{hrs}}{12})+45 $$

2. Now we need to apply the chain rule to determine the derivative \(\frac{d}{d\text{hrs}}\).

    The chain rule is as follows:

$$ \frac{d}{dx}\sin(u)=\cos(u)\frac{du}{dx} $$

    In our case, \(u\) is equal to \(-\frac{\pi \cdot \text{hrs}}{12}\).

    And:

$$ \frac{du}{d\text{hrs}}=-\frac{\pi }{12} $$

    Which gives us:

$$ f^{\prime }(\text{hrs})=36\cos (-\frac{\pi \cdot \text{hrs}}{12}) * (-\frac{\pi }{12}) $$

    Some more simplication:

$$ 36\cdot (-\frac{\pi }{12})=-3\pi $$

3. And, when we account for the fact that \(\cos (-x)=\cos (x)\), we get:

$$ \mathbf{-3\pi }\cos \mathbf{(}\frac{\mathbf{\pi * }\text{hrs}}{\mathbf{12}}\mathbf{)} $$