Let P(x)=F(x)G(x) and Q(x)=F(x)G(x), where F and G are the functions whose graphs are shown.a) Find P′(2)b) Find Q′(7)

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Roosevelt Houghton · Accepted Answer

Step 1a) P′(2) will be the derivative of P(x) evaluated at x=2. So first, we take a derivative.In this section, you should have learned product rule, so P′(x) will look like this:P′(x)=F(x)G′(x)+F′(x)G(x)Now we let x=2P′(2)=F(2)G′(2)+F′(2)G(2)Read these values off the graph.We see that F(2)=3 (When its x-value is 2, the curve F has a y-value of 3)We see that G(2)=2.F′(2) will be the slope of the curve F at 2. A tangent line at F(2) would be horizontal, so F′(2)=0.G(2) has a slope of 12, so G′(2)=12Now we substitute these values into P′(2) to solve:P′(2)=(3)(12)+(0)(2)=(32)+(0)=32Step 2We&#039;re playing by the same rules as above-differentiate the equation, read the values off the graph, substitute the values into the function, and solve.Differentiate (Quotient Rule):Q′(x)=G(x)F′(x)−F(x)G′(x)G(x)2Read the values off the graph:G(7)=1G′(7)=−23F(7)=5F′(7)=14Substitute:Q′(7)=[1⋅(14)−5⋅(−23)][12]Q′(7)=[(14)+(103)]Q′(7)=(4312)=3.5833

Let P(x)=F(x)G(x) and Q(x)=\frac{F(x)}{G(x)}, where F and G are the functions whose graphs are shown. a) Find P'(2) b) Find Q'(7)

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