Find the first partial derivatives of f(x,y,z) = z * arctan (y/x) :at the point (1, 1, 5).
I need serious help . i
So you for first partial in z, you take the derivative as if everything except for z is a variable.
since:
$\displaystyle f(x,y,z)=z* arctan(\frac{y}{x})$
then if z is the only varible, everything attached to it is just some number so you can view the arctan as a constant, say b.
so:
$\displaystyle f(x,y,z)=z*b$
and the partial derivative of that in terms of z then is:
$\displaystyle f_z(x,y,z)=b$
we substitute b back in and get:
$\displaystyle f_z(x,y,z)=arctan(\frac{y}{x})$
can you do the rest from there?
you just take the derivative of arctangent like normal and the other "variables" you just consider them to be constants so like:
what's the derivative of:
arctangent (2x)
$\displaystyle \frac{d}{dx}(arctan (2x)=\frac{1}{1+(2x)^2}*\frac{d}{dx}(2x)=\frac{2}{ 1+4x^2}$
So you can treat the other variables like that 2, they're constants, just that you don't know their value.