5

Point) Let F(z) = f(f(c)) ad G(z) = (F(z))? ad suppose thatf(4) = 3, f(3) = 2, f'(3) = 10, r(4) = 2Find F"(4) and G' (4)F"(4) G(4)...

Question

Point) Let F(z) = f(f(c)) ad G(z) = (F(z))? ad suppose thatf(4) = 3, f(3) = 2, f'(3) = 10, r(4) = 2Find F"(4) and G' (4)F"(4) G(4)

point) Let F(z) = f(f(c)) ad G(z) = (F(z))? ad suppose that f(4) = 3, f(3) = 2, f'(3) = 10, r(4) = 2 Find F"(4) and G' (4) F"(4) G(4)



Answers

Suppose $F(x)=g(f(x))$ and $f(2)=3, f^{\prime}(2)=-3, g(3)=5,$ and $g^{\prime}(3)=4 .$ Find $F^{\prime}(2)$

This is a problem. Number of 53 in which were too find affects open toe the given condition off minus three equal to one and after equal to to one more thing is given that it is a official linear function. So linear function can be written as effects equal toe x plus b Okay. And this means that the function passes through minus three comma one and three common to so slope off the function will be equal toe tu minus one. That is this to minus one Divide by three minus minus three. That is blustery. That is one by six. Okay, so slope is won by six and slope off our this effects slope off effects is always equal toe A in this case, so we have equal to one by six. So let us plug in effects will be equal to one by six x plus b no, This passes through any of the pines. My street number 13 comma two. So let us plug in three comma two here. So to equal to one by six into three plus b 32 times. So we will be called toe to minus one by two, that is three by two. So be is revert to So the question becomes affects. Tickle toe one by six X plus three by two. Oh, Okay. So this is an answer. This is the required linear function. Thank you.

This is Problem number 54 in which we need to find a linear function effects satisfying the condition F minus two equal to seven and F four equal to minus two. So, uh, if you have Alain Eric question, which passes through minus two, Common seven. And for comma minus two. Okay, for call minus two. So we re supposed a linear function. Have x equal toe a X plus B. It's slow for Let us see em will be equal to a No. Uh, since this equation passes through these to find, So according to our formula, the slope will be equal. Do? Why? To minus why? One minus two minus when the *** to minus X one for minus, minus two. That is full plus two minus nine by six. That is minus three way too. Yeah. So now these two things are the same, so he will be equal toe minus three by two. Now let's plug in. Uh, in the function effects will be minus three by two. Plus B. Okay, now it passes through. It passes through, uh, either minus two comma. Seven or four. Cover minus two. Anything we can take. Let us C minus to number seven. So let us plug in here. Seven will be equal to minus three by two into minus two, plus B. So we will be equal toe. Keep less. That is full. So let us play it back. One of my three were to expire Street Twix. So effects will be minus three way to works plus four. So the

All right, So and this question were given sees a function of x and y, which in turn X is a function of T and y is a function of t. All right, And then we're giving a whole bunch of old list of, uh, given so g of threes. He puts two h of three sequence seven G, private three easy with five h private 34 f sub x of to cover 76 f sub y, uh, to call us seven, baby. And then what were asked to determine is the derivative of Z with respect to T when t is equal to two. So basically, this is what we're asked to determine. All right, so, uh, first of all instead of C and when I put ah f x cover, the reason for that is we can see above the given that Z is equal to half of X com away. All right, so now again, we write our change room. So this is a chain rule. So the derivative of F f with respect to x times, a derivative of X and respect ity, plus the derivative of both with respect to lie times the derivative of Fly with Respected So now the name of the game as we're just gonna substitute So instead of X, we're gonna put g of three And instead of why we're gonna put each of three So we plug in, it's right here. So wherever there's an X, we're gonna put you three. Where there's an X, our ally in the Hume aerator on the top we're gonna put you three and wherever there's a wire in a put So now what do we noticed? We know that the definition of eps of axes just a derivative of f with respect to X and that a so this disturbed right here. This ray is this right? Because genius three years to age of 37 and we know the derivative of would respect exit just f sub x. All right, I'm now this right here. Do your Geo three by DT is just g prime. If reaches in derivatives, org promises interpretive of key with respect to t because it only depends on T and then 0.3. All right now again, this right here is the derivative of F with respect alloy at the point g of three years to age of 37 start to calm a seven. So it's just f sub y at two comma 70. This is very good. And in this just HK crown since derivative of HF three derivative of H with respect, idiot. All right. And remember in the top the list of the Given All of these stuff are given so FFC, except to cover 36 g private three is five f supply it to call for seven instead of eight. An h private three is negative for so 16 in spite of its 30 negative eight times negative floors. Positive. 32 So 30 plus 32 is 60. So, uh, the derivative of scene with respected t t close the three year 62.

This is this problem wants us to find the sum of f N G. I have to. So it really translated that this means we need to find Apple too. And And that g of two. Well, if you look above, we can see the f of two is just seven and g of twos. Negative one. So if we add this too, just get the value six. So f plus g of two or the sum of F N g of two is just gonna be the value six. Thank you.


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