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

  • Thread starter MrGoodyear812
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In summary, the conversation is about finding the derivative of y = cos^(x+7)(x) and the process of solving the problem involves taking the natural logarithm of both sides and then taking the derivative to solve for dy/dx.
  • #1
MrGoodyear812
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Help! I haven't the slightest clue on how to do this...

thanks in advance!
 
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  • #2
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.
 
  • #3
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
 
  • #4
Don't worry about 'getting rid of the ln' yet. The problem is about taking derivatives, so take one and see what happens.
 
  • #5
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.
 

1. What is the derivative of cos(x)^(x+7)?

The derivative of cos(x)^(x+7) is (-sin(x)^(x+7))*(ln(cos(x))*(x+7) + (x+7)*sin(x)/cos(x)).

2. How do you calculate the derivative of cos(x)^(x+7)?

To calculate the derivative of cos(x)^(x+7), you can use the power rule and the chain rule. First, rewrite cos(x)^(x+7) as e^(ln(cos(x)^(x+7))). Then, use the power rule to find the derivative of ln(cos(x)^(x+7)), and finally, apply the chain rule to the overall expression.

3. What is the significance of the derivative of cos(x)^(x+7)?

The derivative of cos(x)^(x+7) represents the rate of change of the function at any given point. It can be used to find the slope of the tangent line to the graph of the function and to determine the maximum and minimum values of the function.

4. Can the derivative of cos(x)^(x+7) be simplified further?

Yes, the derivative of cos(x)^(x+7) can be simplified by using trigonometric identities and properties of logarithms. However, the simplified form may not always be more useful or meaningful than the original expression.

5. How does the derivative of cos(x)^(x+7) compare to the derivative of sin(x)^(x+7)?

The derivatives of cos(x)^(x+7) and sin(x)^(x+7) are very similar, with the main difference being the negative sign in front of the sin(x) term in the derivative of cos(x)^(x+7). This is due to the negative derivative of cos(x) with respect to x compared to the positive derivative of sin(x) with respect to x.

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