Having a hard time with a derivative

In summary, the conversation discusses finding the value of dx/dy at (pi/4, pi/4) when x^sin y= y^cos x. There are different methods suggested, including using logarithm properties, but the final answer obtained by implicit differentiation is 4/pi + ln(pi/4) divided by 4/pi - ln(pi/4). The value inside the parentheses signifies the values of x and y being plugged in.
  • #1
Rolando Valdez
4
0
Can not seem to find the answer. If x^sin y= y^cos x

Find dx/dy (pi/4 , pi/4).
If someone could help it would be great.
 
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  • #2
Why don't you use logarithms properties to do it?
 
  • #3
I've tried and came out with x'=-1.07
Can you check this answer.
 
  • #4
wat does trhe value iun the bracket signify if i know that i could help
 
  • #5
the values are (3.14/4, 3.14/4) or (pie/4, pie/4)
 
  • #6
Rolando, I get a different answer. Did you get:

[tex]\frac{dx}{dy}=\frac{\frac{Cos(x)}{y}-Ln(x)Cos(y)}{\frac{Sin(y)}{x}+Ln(y)Sin(x)}[/tex]

When I plug in x=pi/4 and y=pi/4 I get 1.4683
 
  • #7
[tex]x^{sin(y)} = y^{cos(x)}[/tex]

is equivalent to

[tex] \frac{ln(x)}{cos(x)} = \frac{ln(y)}{sin(y)}[/tex]

Implicitly differentiating, I got

[tex]\frac{\dy}{\dx} = \frac {sin^{2}(y) \left( \frac{cos(x)}{x} + sin(x)ln(x) \right) } {cos^{2}(x) \left( \frac{sin(y)}{y} - cos(y)ln(y) \right) } [/tex]

which, at (pi/4, pi/4), is

[tex]= \frac{ \frac{4}{\pi} + ln \left( \frac{\pi}{4} \right) }{\frac{4}{\pi} - ln \left( \frac{\pi}{4} \right)}[/tex]

I don't have a calculator to approximate, though.
 

1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point.

2. Why is it important to understand derivatives?

Derivatives have many practical applications in fields such as physics, engineering, economics, and more. They are used to analyze and optimize functions, model real-life situations, and make predictions.

3. What are the different methods for finding derivatives?

There are several methods for finding derivatives, including the power rule, product rule, quotient rule, chain rule, and implicit differentiation. Each method is useful for different types of functions and can be applied depending on the given problem.

4. How can I improve my understanding of derivatives?

To improve your understanding of derivatives, it is important to practice and work through various problems. You can also seek help from a tutor or online resources, such as videos and practice quizzes. It is also helpful to understand the underlying concepts and connections between derivatives and other mathematical concepts.

5. What are common mistakes to avoid when working with derivatives?

Common mistakes in working with derivatives include incorrect application of the rules, forgetting to use the chain rule when necessary, and not simplifying the final answer. It is important to double-check your work and make sure you are following the correct steps for finding derivatives.

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