Product Rule for Derivatives. Example Problem

Product Rule for Derivatives. Example Problem The function h of x equals 2x squared times f(x), where certain predefined values of f aregiven for x equaling 3. The question tells us to find h prime of 3, and the word "product" in the label tips us off tothe fact that we must multiply two variables together. Previously, if I took the derivative of 3x to the fifth and used the product rule, technically Iwas multiplying two things together. But I didn't need to do that. To find the derivative of a product, such as 3(x - 1), you must use the product rule. This isbecause the derivative of a constant, such as 3, is 0, while the derivative of two variablesmultiplied together is another variable. Recall that if I take the derivative of the product of some function f, and some otherfunction g, then I get something called f g prime plus f prime g. So arbitrarily, I'm going tocall one of these functions f. I typically go by the initials. I'll call this one f. And I'll call this one g. Although it's not that big of a deal to me, I'll still get over the fact that I'm calling f of x ginstead of f. If you find it difficult to follow the formula, simply memorize the steps and write down theformula as you go. I have found that labeling the functions f, f prime, g, and g prime can be an effectivestrategy for my students. It is helpful to read the problem first and then use the formula as a guideline for evaluatingthe derivative. This approach will be especially useful as the problems become morechallenging. So if we call 2x squared, then 4x is the derivative of 2x squared.  And f of x is g , and thederivative of f of x is fprime of x. Then, if you have written the words in that order, f is paired with g prime. The two outermost pieces are f and g. The innermost pieces are paired as follows: f with g,and g with h. Now you can simply write down what you see. The derivative of h prime of x-- again, make sure I use labels-- is equal to 2x squaredtimes f prime of x plus 4x times f of x. Thus, I found h prime of 3 to be 2 times 3 squared times f prime of 3, or −5. Let's color-code this. I'll use purple.Times negative 5 plus 4 times 3 times f of 3.f of 3 is4.So now that I have that, I can scroll down, do the math and find h prime of 3 is negative42.