Partial derivatives of the dependent variable.

Question

I have a function of the form $$f(x,y)=\frac{1}{\exp((y-\mu)/\tau)+1 }$$ where $\mu,\tau,m$ are constants. Also $y^2=x^2+m^2$. How do I calculate $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. Can I just not substitute the value of y in the expression?

Answer 1

If $\mu$, $\tau$,and $m$ are really constants, then $f$ is purely a function of $y$, i.e.

$$f(x,y)=f(y)=\frac{1}{\exp((y-\mu)/\tau)+1 }= \frac{1}{exp((\sqrt{x^2 + m^2} - \mu ) / \tau ) -1}$$

The partial derivatives are then just the regular derivatives, so you just differentiate as you would normally.