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# Two questions watch

1. Why does implicit differentiation work? That is why is ?

And this
http://en.wikipedia.org/wiki/Vertical_line_test
Does that mean that is a function but and are not?
2. (Original post by gangsta316)
Why does implicit differentiation work? That is why is ?
Because the dy terms cancel. >_><_<>_>

http://en.wikipedia.org/wiki/Vertical_line_test
Does that mean that is a function but and are not?
Correct. Well, they're not functions of y, any way.
3. 1. Consider the chain rule.

2. They are not functions (where the definition of function requires one-one mapping).
4. 1) It's a result of the chain rule. Write

z = f(y)

Now we're trying to find dz/dx, which we know is

dz/dx = dz/dy * dy/dx (chain rule)

But dz/dy = f'(y), so

dz/dx = f'(y) * dy/dx, as required.

2) Haven't read the article but y = x^2 is a many-to-one function (i.e. many values can lead to the same answer, e.g. (-5)^2 = 5^2 = 25). x = y^2 is not a function of y and neither is the other one.

If one value of x can lead to more than one value of y, then it's not a function.

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