1. ## Logarithmic Differentiation

Hi, I'm having a bit of trouble with my homework in some parts:

1)

Find y' if:

x^y = y^x

I have no idea on how to even start this.

As for the other:

2)

Use logarithmic differentiation:

sqrt(x)*(e^(x^2))*((x^2)+1)^10

My work:

log both sides:

log y = log(sqrt(x)*(e^(x^2))*((x^2)+1)^10

1/y * dy/dx = 1/2 * ln x + ln e^(x^2) + 10ln((x^2)+1)

Then I am confused..

Thank-you very much!

2. Originally Posted by Prim3
Hi, I'm having a bit of trouble with my homework in some parts:

1)

Find y' if:

x^y = y^x

I have no idea on how to even start this.
Take logs:

$y \ln(x)=x \ln(y)$

Now differentiate with respect to $x$ :

$\frac{dy}{dx} \ln(x) + \frac{y}{x} = \ln(y) + \frac{x}{y}\frac{dy}{dx}$

Now simplify and rearrange:

$\frac{dy}{dx}=\frac{\ln(y)-y/x}{\ln(x)-x/y}$

CB