Derivatives from a graph Let F = f + g and G = 3f - g, where t...

Question

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.

F'(2)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The graphs of F and G are shown. Find H of 3 if H equals F minus G. Awesome. So it appears for this particular problem we're asked to figure out what H3 is if H equals F minus G, and also considering our graph that is provided to us, that shows our different graphs for the curves for F and G. So looking at our graph, we have two separate curves that appear to be V-shaped. And we have our curve in blue, which denotes our Y equals F of X curve, so that's our graph for F. And in red, for our red curve, we have Y equals G of X and Y equals G of X denotes our curve for G, the graph of G. Awesome. So as we could see from these two graphs, considering them, we need to recall a note that the problem itself tells us that H is equal to F minus G, and we can also go ahead and note. That H, which denotes the derivative, the first derivative to be precise. So H equals F minus G. So we need to note that referring to our graph that is provided to us by the problem itself, we need to note that the graphs of F and G both consist of line segments, and the slope of the line tangent to any point on the line, as we should recall a note, will be the slope of the line itself. So we need to consider and recall a note that the graph of F is particularly at, and let's make a note that our graph for F, and let's write this in blue. So considering our graph of F, particularly at 1, is less than or equal to X and X is less than or equal to 4. As this will be the interval where X is equal to 3 is located, and it has an increase of 3 units in Y per unit of increase in X, and this is once again for our function of F, which I'll put a little quotation marks around F. Awesome. So thus the slope will be 3. So note that M denotes slope and M equals 3 in this particular case. So since the derivative of the function at a point is the slope of the line tangent to that point, it can be concluded that F of 3 will be equal to 3, fantastic. So now, our next step is we need to consider the graph of G, which once again is represented in red for our curve on our graph. So considering our graph of G, which is particularly at 0 is less than or equal to X and X is less than or equal to 4. As this is the interval where, and we need to note that this is where X equals 3 is located, and it will have an increase of 1 unit in Y per unit increase in X, and note that this is for our case for G. Awesome. So thus the slope is 1. So note that M equals 1 in this particular case. So similarly, using the slope of G, it could be concluded that G of 3 is equal to 1. So therefore, without a doubt, we could write that H. Of 3 is equal to F 3 minus G 3. So when we plug in our different values for F of 3 and G 3 into our expression, we will find that H of 3 is equal to 3 minus 1, which is equal to 2, and that is our final answer. Hooray, we did it. So 2 is our value for H of 3. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives. <IMAGE>
F'(2)

Key Concepts

Derivative

Sum and Difference of Functions

Evaluating Derivatives at Specific Points

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