How do you differentiate #y=x^2y-y^2-xy#?

Answer 1

#(dy)/(dx)=(2xy-y)/(1+x-x^2+2y)#

Implicit differentiation is used when a function, here #y#, is not explicitly written in terms #x#. One will need the formula for differentiation of product & ratios of functions as well as chain formula to solve the given function.
For this, we take differential of both sides of the function #y=x^2y-y^2-xy# and differentiating we get
#(dy)/(dx)=d/(dx)(x^2y)-2y(dy)/(dx)-d/(dx)(xy)#
#(dy)/(dx)=2xy+x^2(dy)/(dx)-2y(dy)/(dx)-y-x(dy)/(dx)#
Transposing terms containing #(dy)/(dx)# to LHS we get
#(dy)/(dx)-x^2(dy)/(dx)+2y(dy)/(dx)+x(dy)/(dx)=2xy-y# or
#(dy)/(dx)(1-x^2+2y+x)=2xy-y# or
#(dy)/(dx)=(2xy-y)/(1+x-x^2+2y)#
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Answer 2

Simplify then differentiate to find:

#(dy)/(dx) = 2x-1# when #y != 0#

Multiply by #y/(x^2-x-1)# to cover the case #y=0# and find:

#(dy)/(dx) = ((2x-1)y)/(x^2-x-1)#

graph{y=x^2y-y^2-xy [-10, 10, -5, 5]}

Notice that all of the terms are divisible by #y#, so do that first to find:
#1 = x^2-y-x#
with exclusion #y != 0#

We can rearrange this as:

#y = x^2-x-1#

Hence:

#(dy)/(dx) = 2x-1#
with exclusion #y=0#
What happens in the case #y=0#?
The original equation is satisfied, so its graph consists of the parabola #y = x^2-x-1# together with the x-axis #y=0# for which the derivative is #0#.
So to cover the case #y=0# we can multiply #2x-1# by #y/(x^2-x-1)#, since this has value #1# for all points on the curve where #y != 0# and value #0# when #y=0# (and #x^2-x-1 != 0#).
So: #(dy)/(dx) = ((2x-1)y)/(x^2-x-1)#
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Answer 3

To differentiate y=x^2y-y^2-xy, you would use the product rule and the chain rule. The derivative with respect to x is:

dy/dx = 2xy + x^2(dy/dx) - 2y(dy/dx) - y^2 - x(dy/dx)

Then, you would solve for (dy/dx) to find the derivative of y with respect to x.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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