I am working with real numbers and trying to express u as function of x, given the equation:
$x^2-u\,e^u = 4$.
I am a bit lost because of the exponential part. I read something about the Lambert function, so I tried to use it and I have:
$W(u\,e^u) = W(-4+x^2) \implies u = W(-4+x^2)$
Is that correct? If it is, how do i find the derivative with respect to x? Thanks anyone for the help.
$$x^2-ue^u=4$$ $$ue^u=-4+x^2$$ $$u=W(-4+x^2)$$
The derivative of Lambert W is: $$\frac{d}{dx}W(x)=\frac{W(x)}{x(1+W(x))},$$ the derivative of $-4+x^2$ is: $$\frac{d}{dx}-4+x^2=2x.$$
Applying the quotient rule , we get $$\frac{d}{dx}W(-4+x^2)=\frac{2xW(x^2-4)}{(x^2-4)(1+W(x^2-4))}.$$
We also could use the formula for implicit differentiation:
$$x^2-ue^u=4$$ $$x^2-ue^u-4=0$$ $$\frac{d}{dx}u(x)=-\frac{2x}{-e^u-ue^u}$$