Question

f(x)=In(cos^5(3^4)) find dy/dx

Answer 1

Was the three inside the parenthesis supposed to be x?

Answer 2

f(x)=In(cos^5(3x^4)) find dy/dx

Answer 3

\[f(x)= \ln (\cos ^{5}(3x^{4})?\]

Answer 4

First bring the 5 down in front of the log as that is a property of logs. Then take the derivative of the outside function and continue inward using the chain rule

The first step will look like this

let u = (cos^5(3x^4))

d/du(5(ln(u)) = 5/u

Answer 5

So it will be (5/((cos^5(3x^4)))(d/dx((cos^5(3x^4)))

Answer 6

Just keep taking the derivatives using the chain rule :)

Answer 7

If I rearrange it like this it is probably easier to read

(d/dx((cos^5(3x^4))) (5/((cos^5(3x^4))))

Answer 8

Oops with out the cos to the power of five

Answer 9

Like this (d/dx((cos(3x^4))) (5/((cos(3x^4))))

Answer 10

\[f'(x) = \frac{ 1 }{ \cos^5 (3x^4) } (5\cos^4(3x^4)) (12x^3)\]
Finally, you'll get
\[f'(x) = \frac{ 60x^3 }{ \cos(3x^4) }\]
That happened upon applying rules in division of algebraic expression. :)

Answer 11

Hmm i got \[-60x^3\sin(3x^4)/(\cos3x^4)\]

Answer 12

Yeah. I forgot to write about the derivative of cos. It should be \[\frac{ -60x^3\sin3x^4 }{ \cos3x^4 }\]
similarly, it will become\[f'(x) = -60x^3\tan3x^4\]

Answer 13

Yep :)

Answer 14

Do you understand how to get this using the chain rule?

Answer 15

not really

Answer 16

Essentially what you are doing is taking the derivatives the outer functions and working inward.

Answer 17

Multiplying the resulting derivatives.

Answer 18

okay thank you :)

Answer 19

Earlier I took the derivative of the outer function which was ln(u) u being (cos^5(3x^4)).
Then I substituted back in for u. From the point you just keep peeling away the layers in a way by evaluating (d/dx((cos^5(3x^4))).

Answer 20

and so on and so forth until you take the derivative of the innermost function.

Answer 21

okay thank you :) ... uhh i hate calculus -_-

Answer 22

Calculus is great it lets you do so many things!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! :)

Answer 23

yah but I'm not good at math

Answer 24

Practice is the best key. :) Remember this, "Geniuses are made, not born." In short, there are no person born to be genius. It's just that they are so exposed into problem solving. :) Just expose yourself there. You'll see.

Answer 25

aww thank you Yttrium :)

Answer 26

That's what I believe. :3