Having a hard time with a derivative

1. Apr 27, 2005

Rolando Valdez

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.

2. Apr 27, 2005

Pyrrhus

Why don't you use logarithms properties to do it?

3. Apr 27, 2005

Rolando Valdez

I've tried and came out with x'=-1.07

4. Apr 28, 2005

abia ubong

wat does trhe value iun the bracket signify if i know that i could help

5. Apr 28, 2005

rolandov00

the values are (3.14/4, 3.14/4) or (pie/4, pie/4)

6. Apr 28, 2005

saltydog

Rolando, I get a different answer. Did you get:

$$\frac{dx}{dy}=\frac{\frac{Cos(x)}{y}-Ln(x)Cos(y)}{\frac{Sin(y)}{x}+Ln(y)Sin(x)}$$

When I plug in x=pi/4 and y=pi/4 I get 1.4683

7. Apr 29, 2005

Hippo

$$x^{sin(y)} = y^{cos(x)}$$

is equivalent to

$$\frac{ln(x)}{cos(x)} = \frac{ln(y)}{sin(y)}$$

Implicitly differentiating, I got

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

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

$$= \frac{ \frac{4}{\pi} + ln \left( \frac{\pi}{4} \right) }{\frac{4}{\pi} - ln \left( \frac{\pi}{4} \right)}$$

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