MathDad72 2 years ago If f(x) = integral (3,x) sqr root (1+t^3) dt, find (f^1)'(0)

$f(x) = \int\limits_{3}^{x} \sqrt{1+t ^{3}} dt$ find $(f ^{-1})'(0)$

$f(x) = \int\limits_{3}^{x} \sqrt{1+x ^{3}} dx$ $f(x) = (1+x^3)^{1/2}$ $f'(x) = \frac{ 1 }{ 2 }(1+x ^{3})^{^{-1/2}} (3x ^{2})$

3. slaaibak

f'(x) = sqrt(1+x^3) so f'(0= 1

is that setting up: putting the 0 from (f^-1)' $0=(1+x^3)^{\frac{ 1 }{ 2 }}$

5. slaaibak

no. so f'(0)= 1 meaning sqrt(1+0^3) = 1

you just want to sent the equation equal to 0?

oops set

8. slaaibak

bro, you asked me to find f'(0) and I found f'(0), what are you on about?

9. slaaibak

you know evaluating a function at zero and setting it equal to zero are two different things?

correct

11. slaaibak

so you gave f(x). I found f'(x) and then I evaluated it at zero (f'(0)) That's the question, isn't it?

Yes we evaluated f'(x) at zero and we ant to put that into $(f ^{-1})'(0) = \frac{ 1 }{ f'(f ^{-1}(0) }$ Correct? Or am I too tired and my brain is done for the day.

$= \frac{ 1 }{ f'(1) }$

14. slaaibak

oh! I completely overlooked the inverse over there, sorry

yeah those lovely inverses

do we need to back track some for the steps done for doing f'(0) or are we good?

17. slaaibak

so yeah, you solve 0 = f(x) then x=3 do you agree? Then f^-1(0) = 3 ? or alternatively, f(3) = 0

ok

19. slaaibak

then f'(3) = sqrt(1+3^3) = sqrt(28)

dang I didn't need to figure that ugly f'(x)?

21. slaaibak

you did. Well, I did. $f'(x) ={d \over dx} \int\limits_{3}^{x} \sqrt{1+t^3}dt = \sqrt{1+x^3}$

22. slaaibak

it's kinda obvious if you think about it. you should get the same answer back, since you are integrating and then differentiating again. BUT there is a chain rule adjustment that has to be made if there are more complex bounds

I wish these derivative inverses made more sense to me. I am pretty confident in my inverse solving. But these inverse derivatives are killing me.

24. slaaibak

haha, just remember the methods/formulas, do it step by step and you'll be fine

Thanks for your help slaaibak. Have a great night.

26. slaaibak

No problem. Thanx, you too.

27. cobra661966

ok can some one work this problem for me from 1st step to last step. I see you plug x in for t. and 3 in for x but what about the (f^-1)(0) what does that fit in thanks