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 (xg(x)) | x=2

Accepted Answer

Welcome back everyone. In this problem, we want to evaluate the derivative of XG X when X equals to using the graph given below. Now notice that our term here XG of X is a product. How can we differentiate our product? Well, we can use the power rule, the product rule, sorry. And by the product rule of differentiation, OK, what it basically tells us is that the derivative. Of a product FFX multiplied by GFXX. Is equal to the derivative of F of X multiplied by G of X plus the derivative of G of X multiplied by F of X. If we apply that to our problem here, where we know we already have G of X and we let X be F of X, OK. Then. At our point here, OK. Where X equals to, that means the derivative of X G of X. Where X equals to. Is going to be equal to the derivative of F of 2 multiplied by G 2 plus the derivative of G 2 multiplied by F of 2. Now, what do we know here? What, what, what do we already know? Well, we know that for X, its derivative is 1, OK? So that means F 2 is also gonna be equal to 1. That's gonna be multiplied by G of 2, OK? Next, we don't know what the derivative of G 2 is, OK, so we just have to write G2 here. But for F of 2, if X equals F of X, then F of 2 must be 2. So from our problem here, all we need to figure out is the values of G2 and G2. Now, let's start with G2. We'll need the slope of our graph Y equals G X because the derivative of a linear function is equal to its slope. Now, if we look at Y equals G X here, we could take two points. So let's use the point where We have 21 on our graph and let's also use the point up here where this is 45, OK. Ben This means that the derivative of GFX, the slope of G of X is gonna be the difference in Y coordinates, 5 minus 1 divided by the difference in X coordinates 4 minus 2, which would eventually be equal to 2. Since our derivative is a constant, that means G 2 then is also going to be equal to 2. What about G of 2? Well, to get G of 2, we just need to go to our graph and find what the value of G is when X equals 2. When X equals 2, G of X is 1. Therefore, G 2 equals 1. Now that we have G 2 and G of 2, we can substitute it into our derivative. Because no, that means the derivative then. Of X G of X. X equals 2. Is going to be equal to one multiplied by one, OK, plus. 2 multiplied by 21 multiplied by 1 is 12 multiplied by 2 is 4, and when we add those, we get 5. Thus, the derivative of X G of X when X equals 2 is equal to 5. 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 (xg(x)) | x=2

Key Concepts

Product Rule

Evaluating Derivatives

Function Notation

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