We'll now discuss how to calculate the derivative of a function that is one function times another function, like x[sup]2[/sup] times sin(x), which I've lovingly called [code]karl[/code], and graphed below. See if you can guess it by calculating the derivative of [code]karl(x)=x^2*sin(x)[/code] with the code [code]derivative(karl)[/code]. It's fine if you can't.

Clearly, it's not as easy as just the product of the derivatives. [br][br]I hate to say it, but to talk about this with you more easily, I need to introduce a little standard notation found in other calculus courses. The notation isn't hard, but it often leads to confusion, so try not to think about it too much. Here it is: the symbol "d/dx" is just shorthand for "the derivative of". So, for example "d/dx x[sup]2[/sup]" just means "the derivative of x[sup]2[/sup]" which you know is 2x from Monkey Rule 1. Not so bad, right? [br][br]With this new notation we can express the Monkey Rule 6 about the product of two functions a bit more easily.[br][br][b]Monkey Rule 6 (AKA "Product Rule")[/b]: Use the following to calculate the derivative of the product of two functions[br][br][math]\frac{d}{dx}f\left(x\right)g\left(x\right)=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)[/math][br][br]Notice, the left side can be translated to plain English as "the derivative of the product of two functions". The right side can also be translated, but it's cumbersome and unsexy to do so, so let's just leave it in equation-form.[br][br]For instance, in the Geogebra app above, f(x) is x[sup]2[/sup] and g(x) is sin(x). So Monkey Rule 6/the Product Rule says the derivative of the product of the two is 2xsin(x)+x[sup]2[/sup]cos(x).[br][br]Before we move on, try a practice one below. The function [code]sylvan(x)[/code] is the product of x^3 and cos(x). Use the Product Rule to calculate the derivative. Then check it with Geogebra by just typing [code]derivative(sylvan)[/code]. The two functions should be [i]identical[/i]. If not, check what Geogebra did, and see what you did wrong.

There is one special case of the Product Rule that's worth mentioning: when one of the two functions is a constant function as in [code]h(x)=5*sin(x)[/code].[br][br]In this case, [code]h[/code] is the product of [code]f(x)=5[/code] and [code]g(x)=sin(x)[/code]. The Product Rule says the derivative of [code]h[/code] will be:[br][br][math]\frac{d}{dx}h\left(x\right)=\frac{d}{dx}\left(f\left(x\right)\cdot g\left(x\right)\right)=f'\left(x\right)\cdot g\left(x\right)+f\left(x\right)\cdot g'\left(x\right)=0\cdot\sin\left(x\right)+5\cdot\cos\left(x\right)=5\cos\left(x\right)[/math][br][br]The thing to notice is that because one of the product functions (in this case [code]f(x)=5[/code]) is a constant, its derivative is 0 and always "kills" one of the two terms on the right of the product rule. The result is that we really only need to deal with the non-constant part as shown by the fact that the above calculation simplifies to just one term. This is summarized in the following special case of Monkey Rule 6.[br][br][b]Monkey Rule 6b (AKA "Constant Product Rule"):[/b] The derivative of a function which is a constant c times a non constant f(x) can be calculated with the following rule[br][br][math]\frac{d}{dx}c\cdot f\left(x\right)=c\cdot f'\left(x\right)[/math][br][br]With this rule, the derivative of [code]h(x)=5*sin(x)[/code] is much easier; just cut right to the last step: [code]h'(x)=5*cos(x)[/code].