Simple partial derivative

Question

I am learning partial derivatives. I am stuck in understanding the concept. I am expressing the question in form of problems below:

Problem1: $z = xy$, $x$ & $y$ are independent of each other.

what is change in $z$, $\Delta z$, at $x = 1, y = 1$, due to small change in $x$, $\Delta x = 0.001$ and a small change in $y$, $\Delta y = 0.001$

Problem2: $z = xy$

Also $y = x$ (i,e $y$ is function of $x$) what is change in $z$, $\Delta z$, at $x = 1, y = 1$ due to small change in $x$, $\Delta x = 0.001$ and a small change in $y$, $\Delta y = 0.001$

Thanks in advance for answers.

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I found the explanation here http://en.wikipedia.org/wiki/Total_derivative#Differentiation_with_indirect_dependencies thanks 5xum

Answer 1

For a function $f$ of two variables, you can use the estimate $$f(x+\Delta x,y+\Delta y)\approx f(x,y) + \left(\frac{\partial f}{\partial x}(x,y)\right)\cdot\Delta x + \left(\frac{\partial f}{\partial y}(x,y)\right)\cdot \Delta y.$$

For a function $f$ of one variable, that simplifies to $$f(x+\Delta x)\approx f(x) + f'(x)\Delta x$$