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

Question

d/dx (5f(x)+3g(x)) |x=1

Accepted Answer

Hi everyone, let's take a look at this practice of problem dealing with derivatives. This problem says to evaluate the following derivative using the graph given below. And we are looking for the derivative with respect to X of the quantity of 8 multiplied by F of X, plus 12 multiplied by G of X in quantity evaluated at X equal to 2. And we're given a graph that has our functions F of X and G of X plotted on it. F of X consists of a piece-wise 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 comprised of a single linear piece, and it passes through the points of -1 -2 and the point of 2.1. Now we're asked to evaluate this derivative using our graph below. And so we need to evaluate the derivative with respect to X of the quantity of 8 multiplied by F of X plus 12 multiplied by G of X in quantity, evaluated at X equal 2. So to evaluate this derivative, we first need to distribute our derivative inside the parentheses using the distributive properties of derivatives. So when I distribute that inside the parentheses, the first term is going to be 8, multiplied by the derivative with respect to X of F of X. Then we'll have plus 12 multiplied by the derivative with respect to X of G of X. And this whole quantity has to be evaluated at X equal to 2. So now we're actually going to evaluate this derivative at X equal to 2. So this is going to be equal to 8. Multiplied by the derivative of F with respect to X, which is going to be F of X, but we need that evaluated at X equal to 2, that means A is going to be multiplied by F of 2. Then we'll have plus 12, and that's multiplied by the derivative of G with respect to X, which is going to be G of X, but again, we need to evaluate that at X2, so 12 is going to be multiplied by the. Of 2. So now all we need to do is use our graph to find the values of our derivatives of F2 and G2. Now, we call for a linear function, that the derivative of a linear function is equal to its slope. So, we can actually use the slope of our lines on our graphs to determine our values of our derivatives. So the first one that we're going to look at is the F of 2. Now, 2 is contained in the second linear segment of F. And so we're gonna need to choose two points on that segment in order to determine our slope. So we're gonna use the points of 0.4, and the points 1.1. That means that F of 2. It's going to be equal to the slope of this line, which is going to be equal to our change in our Y values. So we'll have 1 minus 4. And that is going to be divided by our change in our X values, which is going to be 1 minus 0. And when we evaluate this expression, this is going to be equal to minus 3. Now, we'll do something similar for the 2. So we need to determine the slope of our line of G of X. And here we're going to choose two points, we're gonna choose the point of -1 -2. And the point of 21. And so that means that G of 2. is going to be equal to the slope of the line, and we're gonna have that calculated by the change in our Y values, which in this case is going to be -1, minus -2. And that's going to be divided by or change it are X, which is going to be 2 minus. -1. And when we evaluate this expression, this is going to be equal to 1 divided by 3. So now we can substitute these values back into the expression that we had found earlier. So that means that 8 multiplied by F of 2. Plus 12 multiplied by G of 2. It's going to be equal to 8, multiplied by minus 3. Plus 12 multiplied by 1 divided by 3. And when we evaluate this expression, this is going to be equal to -20. And that is the answer to this question. 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 (5f(x)+3g(x)) |x=1

Key Concepts

Derivative

Sum Rule of Derivatives

Constant Multiple Rule

Watch next