Question

Using L'Hopital Ruleto evaluate
5x(cos 9x-1)/sin 8x-8x

Answer 1

Take the derivative on both numerator and denominator...

Answer 2

Yes i get down to the third derivative and I jsut can not seem to get the answer correct. here 5limx app. 0 [9x sin-5cos9x] / 8 limx app. 0 [-8cos8x]

Answer 3

That is the third derivative

Answer 4

Show me how you derivated the numerator?

Show me the derivative of : x(cos9x - 1)

Show me..

Answer 5

5limx app0 [-9xsin 9x+ cos 9x-1]

Answer 6

Yes now you are doing right..

Answer 7

Now do the derivative of : sin8x - 8x..

Answer 8

now that equals to 0/0making that derivative indeterminate right
so i kept going but i get stuck on the last part

Answer 9

lim x app 0 [8 cos 8x]

Answer 10

You cannot take out 8 as common of it..
sin8x here 8x is an angle

-8x it is just a value..

Answer 11

Do the derivative separately for them:
Tell me what is the derivative of sin8x?

Answer 12

so [ 8 cos 8x-8] is incorrect

Answer 13

Now you are right..

Answer 14

okay i made an errror in typing it in

Answer 15

Now apply the limits..

cos0 = 1
sin0 = 0

Use this to get the answer..

Answer 16

0/0

Answer 17

No you are incorrect..

Answer 18

Numerator is 0 no doubt but denominator has a certain value..
Check it once more..

Answer 19

0/8

Answer 20

No no sorry you are right..
Now take the derivative once more...

Answer 21

ok

Answer 22

so the next derivative is numerator

Answer 23

Yeah tell me the value after derivative..

Answer 24

is 5 lim x app. 0 [-18x cos 9x -18 sin9x] / 4 lim x app 0 [-8 sin 8x] = 0/0

Answer 25

It is big so take your time and do not do it hastly go slowly I can wait..

Answer 26

ok

Answer 27

9*9 = 81 and not 18..

Answer 28

ok sorry yeah error again
5 lim x app. 0[-81x cos 9x -81 sin9x]

Answer 29

Just and idea. After taking the derivative the first time, try to split into 2 fractions. Do so after taking the derivative the second time.

Answer 30

yeah thanks

Answer 31

[-9xsin 9x+ cos 9x-1] = x(-81cos9x) + (-9sin9x) - 9sin9x = -81xcos9x - 18sin9x

I think I got this much the numerator..

Isn't it??

Answer 32

oh is it all wrong

Answer 33

I am wrong or you are wrong??

Answer 34

oh what did you get

Answer 35

-81xcos9x - 18sin9x

Getting it or not?

Answer 36

i am confuesed this is the part i tend to mess up maybe i am wrong i do not know

Answer 37

See we are going at very fast rate..
Let us do it slowly..
Find the derivative of (-9xsin9x) first and tell me what you got?

Answer 38

ok yeah its just that i have it worked out but let me work it out

Answer 39

okay..

Answer 40

-81x cos 9x

Answer 41

You have to use product rule here @blopez ... So you will get one more term with it..

Answer 42

\[\frac{d}{dx}uv = u.\frac{dv}{dx} + v.\frac{du}{dx}\]

Answer 43

I got the answer. 1215/512

Answer 44

yes okay
-81x cos 9x-81

Answer 45

i need help working the problem out not the answer but thanks

Answer 46

@Mr._To thanks for the answer..
I will carry blopez to that answer...

Answer 47

If you can wait for 5 minutes, I'll post my work.

Answer 48

ok

Answer 49

How you got -81??
Just use the formula and check for answer once more how you did it..

Answer 50

so next is sin 9x

Answer 51

because -9*9

Answer 52

or does the sign change because from sin to cos?

Answer 53

i do not understand why it is not [-81x cos 9x -81 sin9x] if i am using the product rule or am i missing something

Answer 54

Tell me what is the value of u?

Compare (-9x)(sin9x) with uv and tell what is u from here?

Answer 55

See, I will show you in more clearer form:
Wait a minute..

Answer 56

-9x is u

Answer 57

And so v is sin9x and now use the formula slowly and solve for u and v..

Answer 58

If you need steps in between, please let me know.

Answer 59

-81 x cos 9x

Answer 60

\[\frac{d}{dx}(-9x)(\sin9x) = (-9x)\frac{d(\sin9x)}{dx} + \sin(9x).\frac{d(-9x)}{dx}\]

Can you now solve this??

Answer 61

You are finding the first term only but you have to solve for other term also..

See I have written above in more understandable form Please solve the above one..

Answer 62

[-81x cos 9x - 18 sinx 9x]

Answer 63

Now you got it..
Getting how it came??

Answer 64

yes sorry it took so long

Answer 65

Now we are left with the other terms in the numerator that are : Cos9x - 1

Can you find the derivative of these two?
tell me what is that..

No need to say sorry I am just helping you and I will remain helping you until you did get it..

Answer 66

okay thanks

Answer 67

Solving it further or not??

Answer 68

yes iam

Answer 69

Take your time..

Answer 70

sin 9x ??

Answer 71

What is derivative of cosx?
I think it is -sinx not sinx

Be careful @blopez

Answer 72

yes it is so it is -sin 9x

Answer 73

\[\frac{d}{dx}\cos(ax) = -a.\sin(ax)\]

Use the above formula...

Answer 74

It is not -sin9x there is one more mistake in this..

Answer 75

ok
-9 sin 9x

Answer 76

Yes..
We found them separately now join all the derivatives to get numerator..
What it came??

Answer 77

here it goes
5 lim x app. 0 [-81x cos 9x-18 sin 9x]

Answer 78

Yes now take the derivative of denominator...

Answer 79

ok
4 lim x app. 0 [-8 cos 8x]

Answer 80

Firstly tell me what was the denominator before derivative??

Answer 81

ok [cos 8x-1]

Answer 82

Have you taken 8 out of limits?

Answer 83

8 lim x app. 0 [cos 8x-1]

Answer 84

Limit x app 0 (8cos8x - 8)

This is the denominator or you can take it as:

8Limit x app 0 (cos8x - 1)..
Getting or not?

Answer 85

Now tell me the derivative of cos8x??

Answer 86

-8sin 8x or just - 8 sin

Answer 87

-8 sin 8x

Answer 88

Yes now you got it..

Answer 89

So now we are left with last derivative ie the third derivative..
So, be calm and take the third derivative of the numerator..
Let us do it...

Answer 90

ok

Answer 91

Firstly slowly take the derivative of (-81x)(cos9x)

\[\frac{d}{dx}(-81x)(\cos9x) = (-81x).\frac{d(\cos9x)}{dx} + (\cos9x).\frac{d(-81x)}{dx}\]

Solve for this...

Answer 92

ok this is what i had worked out before coming here for the numerator
5 lim x app. 0 d/dx [-x cos 9x -sin 9x]
5 limx app. 0[ 9x sin 9x-5 cos 9x]

Answer 93

How you did this??

Answer 94

is it wrong.

Answer 95

Yes totally...

Just go as I am saying otherwise you will confuse yourself..

Answer 96

ok jaja

Answer 97

Jaja means??