 Often, we can obtain functions f of x, g of x, and so on, but are interested in something based off of these functions. For example, sums, products, quotients, and compositions. We can apply the appropriate derivative rules to find information about the derivatives. And it's helpful to remember, undefined, zero, positive, negative. And while this doesn't make for a rousing school chant, it is actually very useful when considering, well, pretty much everything in mathematics. So for example, let's say we have two graphs, but are interested in the derivative of their product function. So since h of x equals f of x, g of x, then we can use our product tool to find an expression for the derivative. And if I want to find the derivative at negative four, I can substitute in those values. Now, because our ability to read the graph accurately is limited, we're actually going to focus on the sign of the values. So let's take a closer look at the graph. So we see that x equals negative four. f of negative four is zero. f prime of negative four, remember that's the slope of the line tangent to the graph. So let's draw that tangent line. And we see the tangent line slopes downward, and so f prime of negative four is negative. g of negative four, because our graph of y equals g of x is below the x axis at this point, g of negative four is negative. If we draw the line tangent to the graph of y equals g of x at negative four, we see that it slopes upward, and so g prime of negative four is positive. So we know that h prime of negative four, well, that's zero times a positive number plus a negative number times a negative number. Now, zero times a positive number will just be zero, and the product of two negatives is positive, and so h prime of negative four is positive. How about h prime of eight? Well, again, that's going to be f of eight times g prime of eight plus f prime of eight times g of eight, and if we look on our graph at x equals eight, we notice the following. f of eight is positive. We'll draw the line tangent to the graph of y equals f of x at x equals eight, and so this tells us that f prime of eight is also positive. g of eight is positive, and g prime of eight, remember that's the slope of the line tangent to the graph, and if we draw our tangent line, we see that it's horizontal, and so g prime of eight is zero. And so h prime of eight, well, that'll be a positive number times zero plus a positive number times a positive number. A positive number times zero is just going to be zero, and the product of two positives is going to be positive, and so h prime of eight is positive. And if we look at h prime of 12, we need to know about f of 12, g prime of 12, f prime of 12, and g of 12, and so at x equals 12, we note that. And so h prime of 12 is a positive number times a negative number plus a negative number times a positive number. Now the product of a positive and negative is going to be a negative number, and so h prime of 12 is going to be the sum of two negative numbers, and if you add two negative numbers, you get a negative number, and so h prime of 12 is negative. We can also consider compositions of functions. So suppose the graph of y equals f of x is shown. If h of x is the log of f of x, we can find the sign of h prime of one and h prime of three. So h prime of x is, which means we want to know something about f of x and f prime of x. So at x equals one, we see that f of one is positive, and if we draw the tangent line at x equals one, we see that it looks like, and that tells us that f prime of one is positive. And so h prime of one is one over a positive number times a positive number so h prime of one is positive. Similarly, at x equals three, we see that f of three is positive. We see that f prime of three is negative, and so h prime of three is one over a positive number times a negative number. So h prime of three is negative.