Question

Differentiate the funtion

Answer 1

from what?

Answer 2

\(\cos^2(e^{\cos^2(t)})\)?

Answer 3

Yes

Answer 4

So we have the chain rule, we will have

First step is to take the derivative of the outside and leave the inside alone, and multiply all of that by the derivative of the inside

\(2\cos(e^{\cos^2(t)})[\cos(e^{\cos^2(t)})]'\)

Answer 5

Then do it again, \(2\cos(e^{\cos^2(t)})[-\sin(e^{\cos^2(t)})][e^{\cos^2(t)}]'\)

Answer 6

can you find the derivative of \(e^{\cos^2(t)}\)?

Answer 7

Isnt that \[e^{\cos^{2}(t)} [-2 cosx sinx]\] ?

Answer 8

correct

Answer 9

So would this be correct?

Answer 10

It said it was wrong :/

Answer 11

Uh.. I though it was e^u times the derivative of u, but removing it didnt work.

Answer 12

Oh you silly boy -_-
You have some t's... and some x's.

Answer 13

LOL. oops...

Answer 14

Thanks xD

Answer 15

But I do have a quesion

Answer 16

lol i've definitely made that mistake before :)

Answer 17

Where did the 2 come from? the one that in the very front.

Answer 18

Let's rewrite our original expression like this, just for some clarity,\[\large\rm \left[\cos\left(e^{\cos^2t}\right)\right]^2\]

Answer 19

The reason I moved the square notation is...
it makes it a little easier to recognize your `outermost function` when you have a composition.

Answer 20

Clearly the outermost operation is the squaring, yes?

Answer 21

Ah. I see

Answer 22

\[\large\rm \frac{d}{dx}\left[\cos\left(e^{\cos^2t}\right)\right]^2\quad=\quad 2\left[\cos\left(e^{\cos^2t}\right)\right]\frac{d}{dx}\left[\cos\left(e^{\cos^2t}\right)\right]\]So ya, that's where the first 2 comes from.
With a big mess of stuff that follows.

Answer 23

Got it. Thanks for explaining :)

Answer 24

Sorry, I had an appointment I had to get to. Glad it all worked out.