Question

Can I differentiate y=y^2/x^2 using the limit definition/difference quotient?

Answer 1

lol just figured it out
y/y^2=1/x^2
y=x^2
y'=2x

Answer 2

what i was saying is that can you do implicit differentiation with the difference quotient.

Answer 3

hmmm - interesting - never thought about this...

Answer 4

asnaseer, I don't' understand the question, can you please reword the problem properly?

Answer 5

think the question is asking something like can you use the definition of differentiation for something like:\[y=e^{xy}\]. i.e. can you use:\[y'=\lim_{h\rightarrow0}(\frac{f(x+h)-f(x)}{h})\]

Answer 6

I think we can, but it would be a bit messy.

Answer 7

For this problem \( y=e^{xy} \) take the logarithm of both sides and apply differetation but take of the variable you are differentiating.

Answer 8

ok, it turns out that we can. if we rewrite our f(x,y) in a manner to get:
f(x,y) = 0
then we can use the chain rule to get:\[f_x(x,y)+f_y(x,y)\frac{dy}{dx}=0\]therefore:\[\frac{dy}{dx}=-\frac{f_x(x,y)}{f_y(x,y)}\]so if we rewrite the example I gave as:\[y - e^{xy}=0\]then we have:\[f(x,y)=y-e^{xy}\]and we can then use:\[f_x(x,y)=\lim_{h\rightarrow0}(\frac{f(x+h,y)-f(x,y)}{h})\]\[f_y(x,y)=\lim_{h\rightarrow0}(\frac{f(x,y+h)-f(x,y)}{h})\]to calculate the implicit derivative.