I am learning the partial derivative and I am stuck on how to proceed with the following expression.
z = √(4 - x^2 - y^2)
the expressions I need to find are Δ_x z,Δ_y z and Δz, which are are intersections in the plane x,y,z, where
Δ_x z = f(x+Δx,y) - f(x,y)
Δ_y z = f(x,y+Δy) - f(x,y)
Δz = f(x+Δx,y+Δy) - f(x,y)
I have been out of school for a few years now and I can't find any explanations on how to simplify addition expression in a radical such as the expresion z above and this is blocking me.
My first instinct would be to leave the expression complete (for example, Δ_x z = √(4 - (x+Δx)^2 - y^2) - √(4 - x^2 - y^2), but in my previous examples, the -f(x,y) part of the expressions would simplify the non-delta expressions and that confuses me.
Thank you for your help.
If you are trying to compute the partial derivative wrt $x$ of $z = \sqrt{4-x^2-y^2}$, given by $$\frac{\partial z}{\partial x} = \lim_{\Delta x \rightarrow 0} \frac{\sqrt{4-(x+\Delta x)^2-y^2} - \sqrt{4-x^2-y^2}}{\Delta x}$$, a good trick is to multiply numerator and denominator by the conjugate expression $\sqrt{4-(x+\Delta x)^2-y^2} + \sqrt{4-x^2-y^2}$ and simplify. This will eliminate the square roots.