What is the implicit derivative of #-2=xy^2-3xy#?

Answer 1

I found: #(dy)/(dx)=(y(3-y))/(x(2y-3))#

The idea is that you need to derive #y# as well as it represents a function of #x#; so, for example to derive #y^2# you get: #2y(dy)/(dx)#! Where the #(dy)/(dx)# bit takes into accont the #x# dependence of #y#:
in your case you'll get: #0=1*y^2+2xy(dy)/(dx)-3y-3x*1(dy)/(dx)# where I used the Product Rule; collect: #(dy)/(dx)#: #(dy)/(dx)[2xy-3x]=3y-y^2# #(dy)/(dx)=(y(3-y))/(x(2y-3))#
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Answer 2

To find the implicit derivative of the equation ( -2 = xy^2 - 3xy ), differentiate both sides of the equation with respect to ( x ) using implicit differentiation. Here's the process:

  1. Differentiate both sides of the equation with respect to ( x ).
  2. Apply the chain rule whenever you differentiate a term involving ( y ).
  3. Solve for ( \frac{dy}{dx} ) after taking the derivative.

Differentiating both sides with respect to ( x ), we get:

[ \frac{d}{dx}(-2) = \frac{d}{dx}(xy^2) - \frac{d}{dx}(3xy) ]

[ 0 = y^2 \frac{dx}{dx} + x \frac{d}{dx}(y^2) - (3y + 3x\frac{dy}{dx}) ]

[ 0 = y^2 + x(2y\frac{dy}{dx}) - (3y + 3x\frac{dy}{dx}) ]

[ 0 = y^2 - 3y + x(2y\frac{dy}{dx} - 3\frac{dy}{dx}) ]

[ 0 = y^2 - 3y + \frac{dy}{dx}(2xy - 3x) ]

[ \frac{dy}{dx}(2xy - 3x) = 3y - y^2 ]

[ \frac{dy}{dx} = \frac{3y - y^2}{2xy - 3x} ]

So, the implicit derivative of the given equation with respect to ( x ) is ( \frac{dy}{dx} = \frac{3y - y^2}{2xy - 3x} ).

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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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