I need to find (dy/dx) for this problem "x^(1/2)+y^(1/2)=1" in two ways:

a) implicit differentiation, and
b) differentiating an explicit formula for y.



Thank you!

First, to do the explicit diffentiation, rearrange the equation in the form y = f(x), and differentiate as normal. You should end up with dy/dx =f'(x), meaning it's a function of x only.

For implicit diffentiation, you set up the equation in the form F(x,y) = 0. In this case you have:

sqrt(x) + sqrt(y) -1 =0.
Differentiate with respect to x, and you get


\frac 1 2 x^{-\frac 12} + \frac 1 2 y^{-\frac 12} \frac {dy} {dx} = 0\\


which can be rearranged to give the answer as a function of both x and y.

Here's a site that works through a few problems that may be of help:

Visual Calculus - Implicit Differentiation (http://archives.math.utk.edu/visual.calculus/3/implicit.7/2.html)