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Math Help - How do I find Dx tan(x)^secx?

  1. #1
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    How do I find Dx tan(x)^secx?

    How would I find the derivative of tan(x)^{secx}
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by circuscircus View Post
    How would I find the derivative of tan(x)^{secx}
    we have some variable as a power...logarithmic differentiation should come to mind
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    Or we can use the well-known trick a=e^{\ln a},\,\forall a>0
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krizalid View Post
    Or we can use the well-known trick a=e^{\ln a},\,\forall a>0
    right (that trick always slips my mind)
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    I don't see the connection between that and my original equation...
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by circuscircus View Post
    I don't see the connection between that and my original equation...
    Krizalid is saying you could notice that ( \tan x )^{ \sec x} = e^{\sec x \ln ( \tan x)}. and find the derivative of that. \left( \frac d{dx}e^u = u'e^u \right)
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    e^{\sec x \ln ( \tan x)}  secxtanx * \frac{1}{tan x} * sec^2u<br />

    so like this?
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    Quote Originally Posted by circuscircus View Post
    e^{\sec x \ln ( \tan x)}  secxtanx * \frac{1}{tan x} * sec^2u<br />

    so like this?
    to find the derivative of \sec x \ln ( \tan x) you need to use the product rule (while simultaneously using the chain rule to deal with the \ln ( \tan x ) part)
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    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?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by circuscircus View Post
    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:  \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 \sec x \frac 1{\tan x} \sec^2 x you could write \frac {\sec^3 x}{\tan x} and you can change the e^{\sec x \ln \tan x} back to it's original form
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