Let's try calculating the derivative of our model of the length of the day on the Northern Vermont University -- Johnson campus from earlier in the book.[br][br]Recall from [url=https://www.geogebra.org/m/ntenkuu6]earlier activities[/url] that the model we came up with is[br][br][math]g(x)=726.16546+200.61101\sin\left(0.01682x-1.32145\right)[/math] [br][br]Let's calculate the derivative using Monkey Rules 0 through 8. [br][br]First, let's recognize that g is the sum of two functions [code]c(x)=726.16546[/code] and [code]s(x)=200.61101*sin(0.01682x-1.32145)[/code]. [url=https://www.geogebra.org/m/x39ys4d7#material/j9bbkxfe]Monkey Rule 3/Addition Rule[/url] says we can calculate the derivative of [code]g[/code] by simply adding the derivatives of [code]c[/code] and [code]s[/code]. Since [code]c[/code] is a constant function [code]c'(x)=0[/code]. [br][br]Now let's take a look at[code] s(x)=200.61101*sin(0.01682x-1.32145)[/code]. First of all, since [code]s[/code] is the product of a constant function with a sine function, [url=https://www.geogebra.org/m/x39ys4d7#material/twkcnhg2]Monkey Rule 6b[/url] says we only need to focus on the sine part. [br][br]Now, the sine part looks a lot like [code]noah[/code] from the previous activity! Indeed, [code]s(x)[/code] is the composite function [code]f(g(x)) [/code]where [code]f(x)=sin(x) [/code]and [code]g(x)=0.01682x-1.32145[/code] each of which are tackled directly by way of Monkey Rules [url=https://www.geogebra.org/m/x39ys4d7#material/jpb7h5vk]0[/url] and [url=https://www.geogebra.org/m/x39ys4d7#material/nwcsfbgw]5[/url]: [code]f'(x)=cos(x[/code]) and [code]g'(x)=0.01682[/code].[br][br]Using Monkey Rule 8/Chain Rule to put it all together, we get:[br][br][math]g'\left(x\right)=0+200.61101\cdot\cos\left(0.01682x-1.32145\right)\cdot0.01682=3.37428\cdot\cos\left(0.01682x-1.32145\right)[/math][br][br][b]Note[/b]: I rounded to 5 decimal places there at the end when I multiplied 200.61101 and 0.01682 to obtain 3.37428.[br][br]Move on to the next activity to put this to some actual use!