Use the given graphs of f and g to find each derivative.

Question

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

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to evaluate the following derivative using the graph given below. We need to find the derivative with respect to X of the quantity F of F of X, evaluated at X equal to 2. On our graph that we were given, we have two functions F of X and G of X plotted on the graph. F of X is a piecewise function consisting of two linear pieces. The first piece begins at -2.4 and goes to the point 0.4. The second linear piece goes from 0.4 to the point of 2.5 minus 3.5. Our other function G of X is linear, and it passes through two points. It passes through the point of -1-2, and the point of 2.1. Now, in order to take the derivative of our function that we are given in the problem, we're going to need to apply the chain rule. So recall for the chain rule that we have the derivative with respect to X of the quantity of U of V of X. And when we take a derivative of this this uh quantity, it's going to be equal to U of VX, and then this gets multiplied by VX. So we're going to use this rule to calculate our derivative. So, taking the derivative with respect to X of the quantity of F of F of X, that will give us F of F of X multiplied by F of X. And this whole quantity is going to be evaluated at X equal to 2. So now evaluating this at X equals 2, we'll just substitute in 2 for X into our function. So this is going to be equal to F of F of 2, and that's going to get multiplied by F of 2. So now we need to use our graph to determine the values for F2 and F2. So, for F of 2, we just simply need to look at our graph. So, we find the place where X is equal to 2, and we determine the value of F of X when X is equal to 2, and looking at our graph, that corresponds to minus 2. So F of 2 is equal to -2. So now we need to determine F of 2, and here we're going to be taking a derivative of a linear function. And so we recall that when we take a derivative of a linear function, its derivative is equal to its slope. So F2 is going to be equal to the slope of our line. Now, here we have two line segments for F of X, and we're going to choose the one on the right, since that is the segment that contains X equal to 2. And so to find the slope, we're gonna need two points. So we're going to choose the point of 0.4, and the point of 1.1. So that means F2 is going to be equal to the slope of that line, which is equal to our change in our Y divided by a change in X. So for a change in our Y values, we will have 1 minus 4. And that is divided by our change in X, which is going to be 1 minus 0. And when we evaluate this expression, this is equal to -3. Now, we also need to evaluate F of F of 2. And when we look at that quantity, We noticed that we will need to substitute in -2 for F of 2. So, F F 2 is equal to F of -2. And that is actually located on the left segment of our function F of X. And so, what we need to do is calculate the slope of that line. So again, we will actually use two points. We will again use the 00.4, and we'll also use the point of minus 1.0. So that means that F of minus 2 is going to be equal to our change in Y, which is going to be 4 minus 0. And that is going to be divided by Are change in X, which is 0 minus -1. And when we evaluate this expression, this is going to be equal to 4. So that means that if we come back up to our function um that we originally had, which is F of F of 2, multiplied by F2. This is going to be equal to 4 multiplied by minus 3. Which is equal to -12. And that is the answer for this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

Use the given graphs of f and g to find each derivative. <IMAGE>
d/dx (f(f(x))) |x=4

Key Concepts

Chain Rule

Derivative

Evaluating Functions

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