5

Prove the differentintion rule for arctan(r). Begin with the eqquation arctan(z); Four Gual verify thatSolve thc equation involving tAngcnt.arctanl?) forthatMrr...

Question

Prove the differentintion rule for arctan(r). Begin with the eqquation arctan(z); Four Gual verify thatSolve thc equation involving tAngcnt.arctanl?) forthatMrr haveequationUse implicit differentiation to differentiate both sides of the equation Vou obtained in (4).Solve forUse an appropriate right triangle diagram Pythagorean identity to eliminate from Fuur auswer to (c), finally obtaining function Continue simplifying until your ADeicl matches the differentiation formula from the C6 Video

Prove the differentintion rule for arctan(r). Begin with the eqquation arctan(z); Four Gual verify that Solve thc equation involving tAngcnt. arctanl?) for that Mrr have equation Use implicit differentiation to differentiate both sides of the equation Vou obtained in (4). Solve for Use an appropriate right triangle diagram Pythagorean identity to eliminate from Fuur auswer to (c), finally obtaining function Continue simplifying until your ADeicl matches the differentiation formula from the C6 Video Assignmient, (Hint; See # 3)



Answers

(a) Use the Product Rule twice to prove that it $ f,g, $ and $ h $ are differentiable. then $ (fgh)' = f'gh + fgh'. + fgh'. $
(b) Taking $ f = g = h $ in part (a), show that

$ \frac {d}{dx}[f(x)]^3=3[f(x)]^2f'(x) $

(c) Use part (b) to differentiate $ y = e^{3x}. $

Part a of the question asks us to prove that F Times g times h if we take its derivative is equal to F prime G H plus F g Prime H plus F G h Prime. So to do that, we're going to need to use the product rule, which I have written over here in terms of U and V so that we don't get it confused with F and G. So we'll need to identify our you, you Prime V and v prime. So are you, we can say is F and that means that the derivative of you is just the derivative of F and V, We can say is just g h. And now if we want the derivative of G H, we have to use the product rule within the product rule to find the derivative of G. H. So we can say that it is G prime times H plus H Prime times G again just rewriting it in the form of the product rule. And so now we can put these all back together into the original products rule. So we have new prime, which is F prime times V, which is G H plus the prime is G prime H plus H prime G times you which is F So if we distribute the F, we get f prime G H plus half G prime h plus Uh huh G h prime. And now we have proved that this is equal to the derivative of f times g times h. And so now for part B, we want to use that statement to prove this statement. But first, we need to set f equals G and equal to H as the problem tells us So now we can rewrite this function here as simply s cute. Since f times g times H is now f times F 10 f So if we want the derivative cubed, we rewrite f prime G becomes f h becomes f f stays, F g prime becomes f prime. Age becomes f and F becomes f g becomes f and H prime becomes f prime. So now we can rewrite this as three f prime because we have our three F primes here, here and here and f squared. And so now we can rewrite this in a more fancy way, as provided by the problem up here. This simply equals three times the derivative of f times the function f of X squared. So we have shown that this statement provided by B is true. And now finally, we can use this new rule that the derivative of F cubed is equal to three times derivative, the derivative of f times F X squared and we can use that to solve. For this new function y equals x to the fourth plus three x cubed plus 17 explicit 82 cube. So our affects is the inside of the function X to the four plus three X cubed plus 17 X plus 82. And now we want to find what half of X Cube is. So we have X to the fore plus three x cubed plus 17 x plus 82 cubed again. And so we know that this here is equal to this What we've proved in part two so we can say three times the derivative of F. Now, if we want to find the derivative of F, we'll need to use the product rule on each of the terms within ffx since we know it's a polynomial again product rule is the derivative of X to the end is equal to end times X to the end minus one. So first, we'll you will use the product rule on the term X to the four. We see that four is R n So we dropped that in front of X and now four minus one is equal to three. So we add the next term and we can see that our end is equal to three. So we drop that down and multiply it by the coefficient. Three to get nine x and three minus one is equal to X squared. And our next term is 17 x and we can see that r n is simply one. So we drop that down and multiplied by the coefficient. 17 we get 17, one minus one is equal to zero, so we would have X to the zero. But we know that anything raised to the zero power is one. So we just have 17. And for our final term 82 we can see that it is a constant and the derivative of a constant is simply zero. So this here is our derivative of F of X, and so Now we can plug that in to what we got in part B. So three times for X cubed plus nine X squared plus 17. And now we simply have the function f f X squared so we can multiply it. Bye x to the fore. Plus three x cubed plus 17 X plus, 82 squared. And now this is our answer for part C of the problem.

This problem. We want to show that the second equation is a solution to the separable differential equation, which is h of why Times D. Y. D X is equal to G of X, and we can use implicit differentiation to do so. So if we take the derivative of both sides with respect to X and using the fact that why is a differential but unspecified function of acts we get on the left side that D H d y times d Y d x by chain rule is equal to d GDX, but each of y and g of X. These are anti derivatives of h of y and G of X. So that means that we get each of why times D y dx is equal to G of x, which is exactly what we have above, so this completes the problem.

Section eight out four Problem 26 were given a motive. Aargh function of two variables X and Y and were asked to find all four partial derivatives Improve that the mixed partials are equal. So first step, let's find the first derivative with respect to X. So when you differentiate this with the exponential, you're gonna get E to the X y and then I have to differentiate X y with respect to X, and that's just going to leave me with a why, Okay, now, to find this second derivative with respect to X so why is a constant the derivative of E to the X y just e to the X y, and then the derivative of X y with respect to X, it's just why So this is why squared you to the ex wife now the derivative with respect to X and then with respect to why this mixed partial is going to be so You take a look at this and you differentiate with respect to why I might have to use the product rule. So the derivative of why times e to the X y hey, plus why and then the derivative of e to the x y with respect to why this e to the x y, Times X and so this is e to the X. Why one plus X y Now let's go different shape with respect to why the first order partial with respect to Why said, that's going to be e to the X Y and then differentiate x y with respect a y and you simply get X. Take the second partial with respect to why so that's going to be X is a constant you to the X y differentiate x y with respect to y, and that's X. So this is X squared E to the X Y, and then the mixed partial Y X. I'm looking at this. First, here's a different J. This with respect to X that's using the product rule. The derivative of X is one times E to the x y plus X, the derivative eat of the X Y Z did the X Y and then different shape that X y with respect to X, and you simply get why so this is e to the X Y one plus X y. And if we've done this correctly, these two mixed partials should be identical

You have F. So have F G H. Prime to be equal. So, so you have F G goods each right? So we use the product rule in the right hand side. So use food ads. Who? That's room in the right and side implies that I have. So there's the first part A. I have if G. H. Trying to be equal to uh F G crime does each plus MG then each time. So this they would be equal to. So you realize that this is F prime G less. F G crime all times H plus MG. H. Friend. So you noticed that this is a chain room? I've seen you. So if you use the distributive property of multiplication over addition and open their brackets, then you have if G H. Prime should be equal to if. Right? She each plus F G. Prime each plus last F G H. Prime hate. We have me able to fools for the first pass then For the 2nd part which is B. So for B. We have we we have from the part A We realized that F G H crime it's equal to F prime. G. H Plus F three prime each less F G H. Prime. So we replace G. And A. So let's replace we please G. And each with with if so then this implies you have F F F. Right? two vehicles to f prime. If if less if if fine if less if if F prime. So you realize that this is the derivative of order tv. So D you have F of eggs. You the eggs and this would give us is prime mites eggs you have F. X. Squared bloods F. Prime. That's F. X squared plus F crime of eggs. So every time they say eggs here you have f. Of eggs squid. So then this implies you have deal T have F. Of eggs Q. Divided by the eggs. And this is going to be equal to me if brian eggs there is yes eggs squid. So has we have been able to prove that this it's equal to that's then for then this facts for the ness. But see we had from B. So from me That is from the 2nd part B. We had the over the eggs of F. Of AIDS He to be equals three F. Of X. To the power tv Air crime of eggs. Which is basic from which is basically a formula for the derivative of a function cube. So giving given why to be equal to X. to the power four plus do we? X cube Plus 17 eggs plus 80 soon. Q. You can let F. Of X. Well let's so our F. Of X. It's equal to eggs. To the powerful Last three Extra Power Cube last 17 X plus 80 soon. So they this implies that F crime of eggs vehicle to four X cube less nine X squared plus. So they seen. So then using the formula we have d divided by the eggs of of X. To the powerful Last three X. Cubed like 17 X. Last 82. He will be equal to theory to me eggs to the car for last three eggs. Q. Plus 17 X-plus 82 squared. They have four xq plus nine my nine X. Squared Glass 17. So we are using this formula we have to be we have this key F. Of X. This is F. Of X. Q. They this this is good here. So then this would be equal to. So this implies that our way I pray our way I pray then we'll be equal to so why? Prime wipe? Right. It's a white brain wouldn't be equal to we you have eggs to the powerful plus we X. Cubed plus 17 X-plus 82 squared. This is great. Do you have full X. Cubed last night? X squared plus 17 add our final answer.


Similar Solved Questions

5 answers
Problem 20.(1 point) Find the maximum rate of change of f(x,y) = In(x? + 3l) at the point (3, 5) and the direction In which t occuraMaximum rate of change:Direction (unit vector) in which it occurs:Note: You can earn partial credit on this problem:
Problem 20. (1 point) Find the maximum rate of change of f(x,y) = In(x? + 3l) at the point (3, 5) and the direction In which t occura Maximum rate of change: Direction (unit vector) in which it occurs: Note: You can earn partial credit on this problem:...
5 answers
Tekjcnsic Mcnj Maiked Jnje_aquie Is MiZACE)= Zm(EccoiMeIMTE OAnmpaMZAcBi -ecaraiM/Com(Fe Jn /ne+Cecta
Tekjcnsic Mcnj Maiked Jnje_ aquie Is MiZACE)= Zm(Eccoi MeIMTE O Anmpa MZAcBi - ecarai M/Com (Fe Jn /ne+ Cecta...
5 answers
Why doesn t & I6e - M-phenyl undego p-hydride elimination?Why does not planar cobalt complex with an ethyl group axial to it undergo beta hydride elimination"
Why doesn t & I6e - M-phenyl undego p-hydride elimination? Why does not planar cobalt complex with an ethyl group axial to it undergo beta hydride elimination"...
5 answers
In 1999,scientists in Israel developed a battery based on the following cell reaction using what was nicknamed "super iron"2KzFeO4(aq) + 3Zn(s) 7 Fe2O3(s) +ZnO(s) + 2KzZnOz (aq)1st attemptFeedbackd See PerioHow many electrons are transferred in the cell reaction?
In 1999,scientists in Israel developed a battery based on the following cell reaction using what was nicknamed "super iron" 2KzFeO4(aq) + 3Zn(s) 7 Fe2O3(s) +ZnO(s) + 2KzZnOz (aq) 1st attempt Feedback d See Perio How many electrons are transferred in the cell reaction?...
5 answers
Still another antacid used in Tums is calcium carbonate.a) Write a balanced equation for the neutralization of HCl with calcium carbonate. (Hint: it is similar to the neutralization of NaHCOs)-b) Using the equation; calculate how many grams of HCI will be neutralized by 1.0 gram of calcium carbonate_
Still another antacid used in Tums is calcium carbonate. a) Write a balanced equation for the neutralization of HCl with calcium carbonate. (Hint: it is similar to the neutralization of NaHCOs)- b) Using the equation; calculate how many grams of HCI will be neutralized by 1.0 gram of calcium carbona...
5 answers
While measuring the side ofa cube; the percentage error incurred was 3%. Using differentials, estimate the percentage error in computing the volume of the cube:990.06%0.099
While measuring the side ofa cube; the percentage error incurred was 3%. Using differentials, estimate the percentage error in computing the volume of the cube: 99 0.06% 0.099...
5 answers
6. What functional group is present in the compound?A amide aldehyde(A) (C)amineester()
6. What functional group is present in the compound? A amide aldehyde (A) (C) amine ester ()...
5 answers
The sum of the first 51 terms of the arithmetic progression whose 2 nd term is 2 and 4 th term is 8 ,(1) 3774(2) 3477(3) 7548(4) 7458
The sum of the first 51 terms of the arithmetic progression whose 2 nd term is 2 and 4 th term is 8 , (1) 3774 (2) 3477 (3) 7548 (4) 7458...
5 answers
Solve the problem Drbruns 0 3 58125 fruit sule; selling "Ehbo teo ot oranges [ for SII 1 1 did Otegreipefruit 1sold tolio
Solve the problem Drbruns 0 3 58125 fruit sule; selling "Ehbo teo ot oranges [ for SII 1 1 did Otegreipefruit 1 sold tolio...
5 answers
Find the part between the Y ^ 2 = x parabola and the line x = y+ 2. Graphically describe your transactions.
Find the part between the Y ^ 2 = x parabola and the line x = y + 2. Graphically describe your transactions....
5 answers
Rishi florist determines Ihe probabilities for Ihe nber of flower deliver each day: aangemens IhevX10 P(X) 0.2 0.2 0 ; 0 > 0, |Then the values of E(X) and V(X) ae: E(X)-0.25 and V(X)-0.7275E(X)-7.8 and V(X)=1.56None of theseE(X)-1.34 and V(X)-0.9244
Rishi florist determines Ihe probabilities for Ihe nber of flower deliver each day: aangemens Ihev X 10 P(X) 0.2 0.2 0 ; 0 > 0, | Then the values of E(X) and V(X) ae: E(X)-0.25 and V(X)-0.7275 E(X)-7.8 and V(X)=1.56 None of these E(X)-1.34 and V(X)-0.9244...
5 answers
Use the graph below to fill in the missing values.12345-112345-1f(x)f(0)=f(0)= f(x)=0, x=f(x)=0, x= f−1(0)=f-1(0)= f−1(x)=0, x=f-1(x)=0, x= Question HelpQuestion 4: Video1Forum Post to forumSubmit QuestionQuestion 4
Use the graph below to fill in the missing values. 12345-112345-1f(x) f(0)=f(0)= f(x)=0, x=f(x)=0, x= f−1(0)=f-1(0)= f−1(x)=0, x=f-1(x)=0, x= Question HelpQuestion 4: Video1 Forum Post to forum Submit QuestionQuestion 4...
5 answers
Soru 1Yanitiniz:X=1X=4X=6X-5Yaniti temizle
Soru 1 Yanitiniz: X=1 X=4 X=6 X-5 Yaniti temizle...
5 answers
CH-H3 CH3CH3 Eanti staggered'gauche staggeredAEEgauche staggeredEanti staggeredEstrain
CH- H3 CH3 CH3 Eanti staggered 'gauche staggered AE Egauche staggered Eanti staggered Estrain...

-- 0.021389--