Question

Find the derivative:
ln((e^z)(cos(z))+4π3))

I got this for my answer, but apparently it is incorrect.
[(e^z)*(-[sin(z)])+cos(z)*(e^z)+12*(pi^2)]/[(e^z)*cos(z)+4*(pi^3)]

Answer 1

this is my answer in more of a equation form

\[[(e^z)*(-[\sin(z)])+\cos(z)*(e^z)+12*(\pi^2)]/[(e^z)*\cos(z)+4*(\pi^3)]\]

Answer 2

you have to know that:

\(\ln(e^x)=x\)

Answer 3

I am trying to find the derivative of all of this as a whole ln[(e^z)(cos(z))+4π3)], so I'm confused

Answer 4

ln((e^z)(cos(z))+4π3))
ln(e^z) +\ln(cos(z))+4π3))
z + \ln(cos(z))+4π3))
z - sin(z)/(cos(z))+4π3)

Answer 5

ln(ab)=ln(a)+ln(b)

then ln(e^z)=z, and that is why I have z+ ...
and then chain rule for ln(cos(z))+4π3)

Answer 6

i subtract sin, because:
d/dx cos(x) = -sin(x)

Answer 7

oh I should have 1 - sin(z)/(cos(z))+4π3)

Answer 8

deriv of z is 1, I didn't write that sorry

Answer 9

I'm not really following that train of thought, but thank you! Can anyone else try to explain?

Answer 10

\[\ln [e^z \left( \cos z+4 \pi^3 \right)]=\ln e^z+\ln \left( \cos z+4 \pi ^3 \right)=z+\ln \left( \cos z+ 4\pi^3 \right)\]

Answer 11

its derivative
\[=z \prime+\frac{ -\sin z~z \prime }{\cos z+4 \pi ^3 }\]

Answer 12

@Hero

Answer 13

So you would like to take the derivative of the entire thing `without first apply log rules`?

Answer 14

It just told me to find the derivative of ln[(e^z)(cos(z))+4π3)] in which the fashion of ln(x)

Answer 15

So you're trying to apply this rule I guess:
\[\large\rm \frac{d}{dx}\ln(\color{#CC0033}{x})=\frac{1}{\color{#CC0033}{x}}\]But we'll also need to chain rule :)
\[\large\rm \frac{d}{dx}\ln\left[\color{#CC0033}{e^z(\cos z)+4π^3)}\right]=\frac{1}{\color{#CC0033}{e^z(\cos z)+4π^3)}}\cdot\frac{d}{dx}\color{#CC0033}{e^z(\cos z)+4π^3)}\]

Answer 16

Ah I messed up my brackets :p woops

Answer 17

\[\large\rm \frac{d}{dx}\ln\left[\color{#CC0033}{(e^z(\cos z)+4π^3)}\right]=\frac{1}{\color{#CC0033}{(e^z(\cos z)+4π^3)}}\cdot\frac{d}{dx}\color{#CC0033}{(e^z(\cos z)+4π^3)}\]

Answer 18

Don't let the 4pi^3 fool you!
It's a fancy fancy constant, dressed up in a tuxedo and all that jazz.
He's trying to play a trick on you.

Answer 19

No power rule for him.
Derivative of a constant is zero.

Answer 20

Here is our setup for product rule:\[\large\rm \frac{d}{dx}(e^z(\cos z)+4π^3)=\color{royalblue}{(e^z)'}(\cos z)+e^z\color{royalblue}{(\cos z)'}+\frac{d}{dx}4\pi^3\]As mentioned before, derivative of a constant is zero,\[\large\rm \frac{d}{dx}(e^z(\cos z)+4π^3)=\color{royalblue}{(e^z)'}(\cos z)+e^z\color{royalblue}{(\cos z)'}+0\]

Answer 21

Oh obviously that operator should be d/dz, my mistake :(

Answer 22

Taking derivatives:\[\large\rm \frac{d}{dz}(e^z(\cos z)+4π^3)=\color{orangered}{(e^z)}(\cos z)+e^z\color{orangered}{(-\sin z)}\]

Answer 23

Ah ok great!
Your answer looks correct besides the 12pi^2 portion :)

Answer 24

Do you understand why that disappeared?

Answer 25

Oh my goodness yes! This makes so much sense now! Thank you!

Answer 26

yay team \c:/