Question

Find the derivative of the following function f(x)=e^(cosx) + (cosx)^e

Answer 1

Oo tricky little problem :3

Remember the derivative of e^x?

Answer 2

Isn't it the same which is e^x

Answer 3

Yes good.
Same idea with our first term but we'll need to also apply the chain rule.\[\Large\bf\sf \frac{d}{dx}e^{\cos x}\quad=\quad e^{\cos x}\color{royalblue}{\frac{d}{dx}\cos x}\]

Answer 4

So we need to take the derivative of the blue part, chain rule is telling us to.

Answer 5

Exponential gives us the same thing back.
And we're multiplying by the derivative of the inner function.

Answer 6

The derivative of cosx??

Answer 7

Inner function being cosx??

Answer 8

yes, the outer function was e^( )
with cosx being inside of that function.

derivative of cosx?

Answer 9

So the first thing I do is write e^cosx times the derivative of cosx??

Answer 10

This is just for the first term.
You don't have to if you don't want.
I like to do my chain rules like that.
But you can do it all at once if you're comfortable with that.\[\Large\bf\sf \left(e^{\cos x}\right)'\quad=\quad e^{\cos x}(\cos x)'\quad=\quad e^{\cos x}(-\sin x)\]

Answer 11

You can jump from that first step there to the third if you're able to do it in your head.
The middle step is just a way to stay organized.

Answer 12

Yah I d like to stay organize just so I know I'm doing it right

Answer 13

Once I do that do I get the derivative for the other (cosx)^e

Answer 14

Yes.
So for that one, since our variable shows up in the `base`, we can simply apply the `power rule`.

Answer 15

Is it e(cosx)^e??

Answer 16

The e comes down, good.
But then the power rule tells us to subtract 1 from the exponent.\[\Large\bf\sf e(\cos x)^{e-1}\]But then we also have to apply that darn chain rule again!! :O

Answer 17

\[\Large\bf\sf e(\cos x)^{e-1}\color{royalblue}{(\cos x)'}\]

Answer 18

Multiplying by the derivative of the inner function.

Answer 19

So because cosx is in ( ) you derive that instead of the e

Answer 20

Well why did I choose to put the cosx in brackets instead of the e?
I guess it's important that you understand that...

Answer 21

Chain rule is kinda tricky :(
You'll need to practice that some more.

Answer 22

I mean that the cosx is considered the inner so you find the derivative of that instead Of the e which is the outside

Answer 23

So would it basically be e^cosx (-sinx)+e(cosx)^e-1 (-sinx) so far

Answer 24

ya looks good.