What is the rate of change of the function f(x,y)= ln( square root( x^2 + y^2)) at (0, square root(e)) in direction towards the origin?
The unit vector would be (0,1), but I have not been able to find the $\displaystyle
\nabla f
$
I thought that I would just plug in $\displaystyle
(0, \sqrt{e})
$ and (0,0) into the equation, but that doesn't work because you can't when I plug in (0,0) I get ln(0) which isn't possible.
Do you know what I am doing wrong?
$\displaystyle f = \ln \sqrt{x^2 + y^2} = \ln (x^2 + y^2)^{1/2} = \frac{1}{2} \ln (x^2 + y^2)$.
Therefore:
$\displaystyle \frac{\partial f}{\partial x} = \frac{x}{x^2 + y^2}$.
$\displaystyle \frac{\partial f}{\partial y} = \frac{y}{x^2 + y^2}$.
Therefore:
$\displaystyle \nabla f = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j = \frac{1}{x^2 + y^2} \, (x i + y j)$.
Evaluate at $\displaystyle (0, \sqrt{e})$.