Here's your derivative from the previous activity. It shows us that the derivative of [code]noah[/code] is [code]noah'(x)=5cos(5x+2)[/code].

I'll be the first to admit that [code]noah[/code] and his derivative aren't the prettiest functions. However, [code]noah[/code] and his derivative offer us an excellent opportunity to see the 8th and final monkey rule. To help us get there, let's think about the functions that are joined by composition to form [code]noah[/code]. To do so, think about [code]noah[/code] like the computer program as he is, and ask yourself: what's the first thing that's done to the input, x, by [code]noah[/code]? [br][br]The answer is, that the first thing that's done to the inputs of [code]noah[/code] is they're multiplied by 5, and then 2 is added to that. So, if we were to break [code]noah[/code] into two phases, say [code]g[/code] and [code]f[/code], we might say phase one is the linear function g[code](x)=5x+2[/code].[br][br]After phase 1, or g as we are calling it, phase 2 is to apply sine to the outputs of g. So phase two is simply the function[code] f(x)=sin(x)[/code]. [br][br]Noah is the composite of phases 1 and 2, first [code]g(x)=5x+2[/code] and then [code]f(x)=sin(x)[/code]. With these phases picked apart, we see that we can really declare [code]noah[/code] as the composite function[br][br][code]noah(x)=f(g(x))[br][/code][br]The reason for picking apart the phases of noah is to now look back at noah'(x) and realize that noah'(x) is a conglomeration of the derivatives of his phases. Indeed, since[code] g(x)=5x+2 and f(x)=sin(x)[/code], we can use earlier Monkey rules to calculate the derivatives [code]g'(x)=5[/code] and [code]f'(x)=cos(x)[/code]. Now you can see that [br][br][code]noah'(x)=5*cos(5x+2)=cos(5x+2)*5=f'(g(x))*g'(x)[/code][br][br]The only reason I switched the order of the multiplication was so that g'(x) would appear at the end of the equation to match the standard phrasing of our final Monkey Rule:[br][br][b]Monkey Rule 8[/b] [b](AKA "Chain Rule"):[/b] The derivative of a composite function can be calculated with the following rule:[br][br][math]\frac{d}{dx}f\left(g\left(x\right)\right)=f'\left(g\left(x\right)\right)g'\left(x\right)[/math][br][br]It's not uncommon to think that this rule "blows up" [code]f(g(x))[/code] in the process of calculating the derivative. This isn't a bad way to think about this, and we'll come back to this idea later when we study integrals.[br][br]For now though, move to the next activity to get a bit of practice with the Chain Rule and the other Monkey Rules.