How do I find the derivative of cos(x)^(x+7)?

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Homework Help Overview

The discussion revolves around finding the derivative of the function \( y = \cos(x)^{(x+7)} \), which involves the application of logarithmic differentiation. Participants express uncertainty about the process and the steps involved in differentiating the function.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of natural logarithms to simplify the differentiation process, with one suggesting taking the logarithm of both sides. Questions arise about how to manipulate the equation after applying the logarithm, particularly regarding the treatment of \( y \) and the implications of differentiating both sides.

Discussion Status

The discussion is ongoing, with participants exploring different aspects of logarithmic differentiation. Some guidance has been offered regarding the differentiation process, but there is no explicit consensus on the next steps or the final approach to take.

Contextual Notes

Participants indicate a lack of clarity on the differentiation process and express confusion about the manipulation of logarithmic expressions. There is an emphasis on understanding the derivative rather than solving the problem outright.

MrGoodyear812
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Help! I haven't the slightest clue on how to do this...

thanks in advance!
 
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MrGoodyear812 said:
Help! I haven't the slightest clue on how to do this...

thanks in advance!

[tex]y=cos^{x+7}(x)[/tex]

Start by taking the natural logarithmic of both sides.
 
even the y?

so i'd have:

ln(y) = (x+7)ln(cos(x))

how do i get rid of the ln on the y?

cause i knew it was a ln problem, just i don't know how to get rid of the ln on the y
 
Don't worry about 'getting rid of the ln' yet. The problem is about taking derivatives, so take one and see what happens.
 
If you take the derivative of the left side, you should get (1/y)(dy/dx). On the right side, you should have the derivative of ((x + 7)ln(cos x)). Solve the resulting equation for dy/dx.
 

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