Solving the partial deriative by definition, what to do with constants?

Question

If I want to find a partial deriative by of y=2x-5+4z-(xz+57) for x, by definition, just like simple deriative, what I should to do with constants? By definition, I mean trought limit

Answer 1

Your function $y$ is a function of two variables, namely $y(x,z)$. The definition of the partial derivative of $y$ with respect to $x$ is given by

$$ \frac{\partial y(x,z)}{\partial x} = \lim_{h \to 0} \frac{y(x+h,z) - y(x,z)}{h} $$

In our case this evaluates to

\begin{align} &\lim_{h \to 0} \frac{2(x+h)-5+4z-((x+h)z + 57) - 2x + 5 -4z +xz + 57}{h} \\ &=\lim_{h \to 0} \frac{(2x + 2h + 4z -xz -hz-62) -(2x+4z-62-xz)}{h}\\ &=\lim_{h \to 0} \frac{(2h-hz) + (2x + 4z -xz-62) -(2x+4z-62-xz)}{h}\\ &=\lim_{h \to 0} \frac{h(2-z)}{h}\\ &=\lim_{h \to 0}2-z = 2-z \end{align}