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.

G'(2)

Accepted Answer

Hello there. Today we're going to 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 K 6 if K is equal to 5 multiplied by F minus 2 multiplied by G. Awesome. So it appears for this particular prong we're trying to figure out the value of K 6 if K is equal to 5 multiplied by F minus 2 multiplied by G, considering that we have two different curves graphed for F and G on. Our graph that's given to us by the problem itself. So we have two curves denoting our graphs for F and G. So our curve in blue denotes Y equals F of X, which is our graph for F, and we have a curve in red which denotes Y equals G of X, which is our graph for G. And as you can see, they are V shaped, so they're shaped like V's. As we can see, and we're going to be using these graphs to help us solve for K of 6, which is our final answer we're ultimately trying to solve for. So let us note that the problem itself tells us that K is equal to 5 multiplied by F minus 2 multiplied by G. And let us also note that we can also write that K where K denotes the first derivative of K and K is equal to 5 multiplied by F minus 2 multiplied by G. So we also need to know and recall looking at our graphs as we can see the graphs for F and G both consist of line segments, and as we should recall and note, the slope of the line tangent to any point on the line will be the slope of the line itself. So considering the graph of Y first, Sorry, not the graph of Y, the graph of F. So considering the graph of F first. Particularly, this is for at 4 is less than or equal to X and X is less than or equal to 10. So, once again, we're considering the graph of F, particularly at the specific interval of 4 is less than or equal to X and X is less than or equal to 10, and this is where the interval X equals 6 is. Being considered. So this is the interval where X equals 6 is located, and it has a decrease unit of, so it has a decrease in a unit of NY per 2 units of increase in X. So I'll say that again since that was a little bit confusing. So Once again, considering our specific interval where it is located at X equals 6, it will have a a decrease of a unit in Y per 2 units of increase in X. So therefore, that will mean that the slope, which the slope is denoted as M, so M will be equal to -12. Awesome. So as a result, due to the fact that there's a decrease of a unit in Y per 2 units in increase of X, there will be a slope at -12, and since the derivative of the function at the point is the slope of the line tangent to that point, as we should recall. It can be concluded that F of 6 will be equal to -12. Fantastic. So similarly, we now need to consider the graph of G. And for G now, considering our graph for G, particularly at the interval where 4 is less than or equal to X and X is less than or equal to 7, as this is the interval where X is equal to 6 is located. It will have a decrease of 2 units in Y per unit of increase in X, which will mean that it's slow for this particular case will be -2. So similarly, we can conclude that G 6 will be equal to -2. So now we can officially solve for K of 6. So K 6 is equal to 5 multiplied by F of 6. Minus 2 multiplied by G of 6, so plugging in our different values for F6 and G6, we will find that K6 is equal to 5 multiplied by negative 1/2 minus 2 multiplied by negative. 2. Which will mean when we plug this into our calculator and we solve for our final answer, we will find that it's equal to 3 halves, and that is our final answer. Hooray, we did it. So, that is it. 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.
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G'(2)

Key Concepts

Derivative

Sum and Difference of Functions

Evaluating Derivatives at Specific Points

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