# How do I find Dx tan(x)^secx?

• Sep 25th 2007, 10:44 AM
circuscircus
How do I find Dx tan(x)^secx?
How would I find the derivative of $\displaystyle tan(x)^{secx}$
• Sep 25th 2007, 11:01 AM
Jhevon
Quote:

Originally Posted by circuscircus
How would I find the derivative of $\displaystyle tan(x)^{secx}$

we have some variable as a power...logarithmic differentiation should come to mind
• Sep 25th 2007, 11:50 AM
Krizalid
Or we can use the well-known trick $\displaystyle a=e^{\ln a},\,\forall a>0$
• Sep 25th 2007, 11:57 AM
Jhevon
Quote:

Originally Posted by Krizalid
Or we can use the well-known trick $\displaystyle a=e^{\ln a},\,\forall a>0$

right :D (that trick always slips my mind)
• Sep 25th 2007, 09:32 PM
circuscircus
I don't see the connection between that and my original equation...
• Sep 25th 2007, 09:54 PM
Jhevon
Quote:

Originally Posted by circuscircus
I don't see the connection between that and my original equation...

Krizalid is saying you could notice that $\displaystyle ( \tan x )^{ \sec x} = e^{\sec x \ln ( \tan x)}$. and find the derivative of that. $\displaystyle \left( \frac d{dx}e^u = u'e^u \right)$
• Sep 25th 2007, 10:40 PM
circuscircus
$\displaystyle e^{\sec x \ln ( \tan x)} secxtanx * \frac{1}{tan x} * sec^2u$

so like this?
• Sep 25th 2007, 10:43 PM
Jhevon
Quote:

Originally Posted by circuscircus
$\displaystyle e^{\sec x \ln ( \tan x)} secxtanx * \frac{1}{tan x} * sec^2u$

so like this?

to find the derivative of $\displaystyle \sec x \ln ( \tan x)$ you need to use the product rule (while simultaneously using the chain rule to deal with the $\displaystyle \ln ( \tan x )$ part)
• Sep 25th 2007, 11:07 PM
circuscircus
$\displaystyle e^{\sec x \ln ( \tan x)} \sec x\tan x \ln ( \tan x) + \sec x \frac{1}{\tan x}\sec^2x$

so would be like this?
• Sep 25th 2007, 11:27 PM
Jhevon
Quote:

Originally Posted by circuscircus
$\displaystyle e^{\sec x \ln ( \tan x)} \sec x\tan x \ln ( \tan x) + \sec x \frac{1}{\tan x}\sec^2x$

so would be like this?

use parentheses! you have the idea, but the answer as written is wrong.

you should have: $\displaystyle \left( \sec x\tan x \ln ( \tan x) + \sec x \frac{1}{\tan x}\sec^2x \right)e^{\sec x \ln ( \tan x)}$

by the way, this can be simplified a lot, for instance, instead of writing $\displaystyle \sec x \frac 1{\tan x} \sec^2 x$ you could write $\displaystyle \frac {\sec^3 x}{\tan x}$ and you can change the $\displaystyle e^{\sec x \ln \tan x}$ back to it's original form