Derivatives from graphs Use the figure to find the following d...

Question

Derivatives from graphs Use the figure to find the following derivatives.

d/dx (f(x)g(x)) | x=4

Accepted Answer

Welcome back, everyone. In this problem, we want to evaluate the derivative of FFX GFX where X equals 3, using the graph given below. Now, how are we going to evaluate our derivative? Notice that this is a product, OK? So we could use the product tool of differentiation. And recall that by the product rule. It basically tells us that the derivative of our product, OK. Product of FFX and GFX. In this case, where X equals 3. Is going to be equal. The the derivative of FFX. Multiplied by geofix. Plus, the derivative of GX multiplied by FX, OK, where X equals 3. And then if we substitute X equals 3 into our derivative, then we're going to get the derivative of F sorry F 3 multiplied by G 3 plus G3 multiplied by F of 3. So what this basically tells us, and this should be an equal sign here. What this is basically telling us is that if we can find the derivatives or the slopes of FF X and the G of X and then substitute 3 into our derivatives along with substituting 3 into the original functions, then we should be able to find the derivative using our graph. So let's go ahead and try to do that. First, let's try to find the slope of FFX using points and F of X. So here I noticed that FFX has the point. 03 and 30. So let's use those points to find the slope of FFX. No, it's in blue, so let me change my color here. No The derivative or the slope of FFX then. Is going to be equal to the change in Y values, that's 0 minus 3 divided by the change in X values, 3 minus 00 minus 3 is -3, 3 minus 0 is 3, and 3 divided by 3 equals -1. Now if the derivative of FF X is -1, then even if we substitute 3 into our derivative, it will still be -1 because our derivative or slope is a constant. Now what about FF3? Let me put this up here actually. What about FF 3? Well, no, OK, let me just make some changes here just to make this look a little bit neater, OK. So no for to find F of 3, all we need to do is to go to our graph and look at when X equals 3. When X equals 3, F of X is 0. Therefore, F of 3. Is gonna be equal to 0, OK? So now we have our values of F3 and F of 3. Next, let's do the same to get the values of G3 and G of 3. Now if we think about it here, so we put next. Then, first, the derivative of G of X, again, we take two points. So notice here, oops, my apologies. Notice here that we could use the points where G is 54. OK. And where it's 21, OK. So, G of X is going to be equal to the difference between Y coordinates, 5 minus 1 divided by the difference between X coordinates, 4 minus 2.5 minus 1 is 44 minus 2 is 2, and 4 divided by 2 equals 2. So if G X equals 2, then that means G 3 is also gonna be equal to 2. Since we have G 3. If we also try to find the value of G of 3, then when X equals 3, if we go to our graph, notice that G equals 3. So G of 3 equals 3. Now that we have F3, F3, G3 and G3, we can plug them into our product tool, OK. So, therefore, OK, that means the derivative, the derivative. Of F of X multiplied by G of X when X equals 3. Is going to be equal. To F of 3, which is -1, multiplied by G of 3, which is 3, plus G 3, OK, which is 2 multiplied by F of 3, which is 0. -1 multiplied by 3 is -3 and 2 multiplied by 0 is 0. Thus, the derivative of FX multiplied by G of X when X equals 3 equals -3. Thanks a lot for watching everyone. I hope this video helped.

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>
d/dx (f(x)g(x)) | x=4

Key Concepts

Product Rule

Evaluating Derivatives at a Point

Graphical Interpretation of Derivatives

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