Implicit Differentiation

[Krishang]

[math]x^{xy} + y^{exy} = xe^y[/math]
I was given this problem to differentiate, wrt x. I am just unsure of the differentiation of the first part. Do I take it as its own seperate expression and then take logarithm and then do the differentiation? I am unsure about this part. Thank you.

 

[blamocur]

The standard approach for integrating [imath]x^x[/imath] is representing it as [imath]x^x = e^{x \ln x}[/imath]. Do you see how to do the rest?

 

[nasi112]

[math]x^{xy} + y^{exy} = xe^y[/math]
I was given this problem to differentiate, wrt x. I am just unsure of the differentiation of the first part. Do I take it as its own seperate expression and then take logarithm and then do the differentiation? I am unsure about this part. Thank you.

We can't help you if you do not tell us what is e. Is it constant or variable? You also did not say what is the independent variable. Even if wrt x means the independent variable is x, you can't isolate it. So I think your equation is wrong or you made it up.

Though I can help you with this example.
[math]y = x^x[/math][math]\ln y = \ln x^x[/math][math]\ln y = x\ln x[/math][math]\frac{1}{y}y' = \ln x + x\frac{1}{x}[/math][math]\frac{1}{y}y' = \ln x + 1[/math][math]y' = y(\ln x + 1)[/math][math]y' = x^x(\ln x + 1)[/math]