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Suppose that f(5) = 1,f'(5) = 6, 9(5) -3,and g'(5) =2Find the values of (fg) (5) and (g /f)(5)....

Question

Suppose that f(5) = 1,f'(5) = 6, 9(5) -3,and g'(5) =2Find the values of (fg) (5) and (g /f)(5).

Suppose that f(5) = 1,f'(5) = 6, 9(5) -3,and g'(5) =2 Find the values of (fg) (5) and (g /f)(5).



Answers

Suppose that $ f(5) = 1, f' (5) = 6, g(5) = -3, $ and $ g'(5) = 2. $ Find the following values.
(a) $ (fg)' (5) $

(b) $ (f/g)' (5) $

(c) $ (g/f)' (5) $

Okay, so here we have the functions F and G are defined by ordered pairs. We have inputs and outputs. So F composed G five. We know that means F of G 05. So we have F of G of five. So what is G five G of five is equal to seven? Since when the input is five, function G the output is seven. So therefore we're finding here F seven and F of seven um is well it's not possible, right, because um looking at function F right, there is no input, there is no input um for seven. So F of seven. Well that's undefined, there is no input, so there is no input of seven in function F. Therefore, um we cannot find um F composed F composed G five. Right? Um because ff seven is undefined, so so we're done so were undermined.

So what I see here is I have two functions multiplied together to the derivative at X equals five. So what this is telling me is I'm doing the product rule between F and G. And if I were to just if I were just ignoring this five here, well, this would be would be the first derivative of F times drew it over the first function times a second function plus the first function times the derivative of the second function. But this is all evaluated at five. So the product rules just each of these terms evaluated at back and thankfully I have all this information here. I know that F prime of 56, G at X equals five. The original Crookshanks, it was five is negative three plus The original f function at five is 1 and the original and the first derivative of GF five is too simplifying this. Multiply before I add negative 18 plus two. This will be negative 16. That's my final answer here. Now let's say it was dividing two functions and taking the first road and we'll have two functions that I'm dividing. So this is the quotient role derivative of the first times the second minus the first times derivative of the second, All over the original bottom to the second power. And recall that this is all evaluated at five. So each of these terms is evaluated at five. I know from above that this will be negative. This first term will be negative 18 minus the original might as positive too. All divided by G at five. The original function G F five is negative three. All raise the second power negative 18 minus two, gets more negative negative 20 negative three. To the second power negative three. Types itself as positive. Nine. -20/9. And then finally, let's say I reverse this quotient rule. Now I'm doing G up top on bottom. Will this change the answer? Well, Let's see, derivative of G at five times The original. The bottom function at five minus. Um The top function at five times the first derivative of FF five all over. Now, the bottom term is F five. Okay, so G prime at five is two times F prime five is one minus. G. Prime of five is negative three times F prime of five is six. All over. The original function. FF five is going to be one to the second power. So this will look different. This will be too -18. All the better by one, which will just be 20, which is different than negative. 20/9.

The question gives us the values of F at five F Prime at five g at five g prime at five and asks us to solve for the derivative of F times, G at five, the derivative of F over G at five and the derivative of G over F five. So to start at a, we can see that the derivative of F times G is just our product rule. And we know that equals F Prime G plus g Prime F. And now if we want the value of five, we need F prime at five times G at five plus g prime at five plus sorry times F at five. And now again, the answer gives her The problem gives us the values of F private five, which is six. We multiply that by g G at five is negative. Three. Add that to G Private five is two and multiply it by f f five is one and we can simplify. We get negative 18 plus two and this is negative 16, which is our answer for per a And now for part E. B, we can see that the derivative of F over G takes the form of the quotient rule, which we know is F prime G minus g prime f over over she squared and again if we want the value at five. We need F prime at five times G at five minus G prime at five times half at five over G F five squared and so we can plug in the values that the problem gives us. F Private five is six times G, at five is negative. Three minus G private. Five is too. Times F F five is one over G at five Squared, which is negative. Three squared. And if we simplify, we get negative 18 minus two over nine and then we can come to our answer. Negative 20/9 and now Part C. We can see that G over F. The derivative is takes the form of the quotient rule, but G and F are switched, so we need to switch the positions of G and F and the quotient rule. So we'll write G over F It's derivative. We will have G prime F minus F prime G over f squared. And if we want the value of five, we need G prime at five times F five minus F prime at five times G at five over F five squared and plugging in. We have G Prime of five is two times F at five is one minus. F prime at five is six times G. At five is negative. Three over F at five again is one, and that's squared so we can simplify to get to minus six times negative. Three is negative, 18 over one squared is one, and we get two plus 18, which equals 20 and that's our answer for part C of the problem.

Yeah. Hello. So here we are. Supposing that F of five is equal to one. F prime of five is equal to six. G five is equal to negative three and G prime of five is going to be equal to two. So four part A. Um We want to take the value F times jeep prime of five. So F times G prime of five. Okay, so first we evaluate the derivative of the product. So F um times G prime of five. That's F five times G prime of five plus G F five times F primer five by the quotient rule by the product will excuse me and buy the product. We have one times two plus native three times six. Right, it's going to be equal to um one times two plus negative three times six. So that's going to be equal to minus 18 which is negative 16. So therefore F times G prime of five is going to be equal to negative 16. And then for part B we find the result F divided by G prime of five. Self apart be here. That's going to be F over G Prime five. So for the coalition role within the bottom times the derivative of the top minus the top times the derivative of the bottom all at the bottom of the bottom square. So that's going to be -3 times six. And then minus one times two. All divided by negative three squared. That's gonna be negative 18 minus 2/9. That's going to be equal to negative 20/9. So therefore we have F Divided by G. Prime 05 is equal to negative 29th. Um And then um we then take the result to value driven of the caution. Um G divided by F. Prime of five. So that's going to be G over F. Mhm. Crime uh five. So that's going to be equal to well one times two and then minus a negative three times six. All divided by one squared. So it's gonna be equal to two plus 18/1, that's 20/1, which is equal to 20. So therefore G over F. Prime or five is going to be equal 2 20 mm.


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