Having a hard time with a derivative

[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.

 

[Pyrrhus]

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

 

[Rolando Valdez]

I've tried and came out with x'=-1.07
Can you check this answer.

 

[abia ubong]

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

 

[rolandov00]

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

 

[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

 

[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.