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

Question

d/dx (g(f(x))) |x=1

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're asked to take the derivative with respect to X of the quantity of G of F of X evaluated at X equal to 2. Below in the graph we're given 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 the point of -2.4 and goes to the point of 0.4. The second piece goes from 0.4 to the point of 2.5 minus 3.5. G of X is also linear on this graph, and it passes through the points of -1-2 and the point 2-1. Now, in order to evaluate our derivative here, we're going to be using the chain rule. So recall for the chain rule that we have the derivative with respect to X of the quantity of U V X. And when we take the derivative of this function, this is going to be equal to U of V of X. Multiplied by VX. So we're gonna use our chain rule here to calculate our derivative. So we're gonna take the derivative with respect to X of the quantity of G of F of X, evaluated X equal to 2, we're going to get G of F of X. Multiplied by F of X. And this quantity is evaluated at X equal to 2. So now substituting in 24 X in our expression, this is going to be equal to. G of F of 2. Multiplied by F of 2. So we'll need to use our graph to determine, to begin with, F2 and F of 2. So we're going to begin with F of 2, and for F of 2, we just simply need to look on the graph. So on our function F of X, when X is equal to 2, F of X is equal to minus 2. So we can just read that directly off of the graph. Now we need to determine the value of F 2. And here recall that when we take a derivative of a linear equation, the derivative is equal to the slope of the line. So here we just need to calculate the slope of a line, and we're actually going to be using the second segment, since that is the segment that contains X equal to 2. So to calculate the slope, we'll just need two points. The first point we're going to use is 0.4, and the second point that we're going to use is 1.1. And so F of 2 is going to be equal to the slope of this line, which is equal to our change in our Y values divided by our change in X. So our change in our Y values is going to be 1 minus 4. And that is divided by our change in our X values, which is 1 minus 0. And when we evaluate this expression, this is equal to -3. Now we need to also look at our expression. We're gonna have to calculate G of F of 2. So, 4 G of F of 2, since we have a value for F2, this is going to be equal to G minus 2. And here we need to calculate the derivative of G when X is equal to -2. And since we just have a single segment here and G is linear, the derivative here is also going to be equal to the slope of the line. And so we're going to choose again, two points. We'll choose the point of -12, and the point of 21 as our points to use to calculate our slope. So we'll have the G of -2. It's going to be equal to our change in our y values, which is going to be -1, minus. -2. And that quantity is going to be divided by our change in X, which in this case is going to be 2 minus -1. And when we evaluate this expression, this is going to be equal to 1 divided by 3. Now we have all the information that we need to evaluate our expression for our derivative. So, coming back to the question that we calculated, which was G of F of 2. Multiplied by F of 2. This is going to be equal to for G of F of 2, that is equal to G minus 2, which is equal to 1 divided by 3. And that gets multiplied by 2, which is minus 3. And so evaluating this expression, this is going to be equal to -1. And that is the answer to 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 (g(f(x))) |x=1

Key Concepts

Chain Rule

Derivative

Evaluating Derivatives at a Point

Watch next