How do you implicitly differentiate # x^2-(x-1)(x - y)^2 = x^2 / y^2#?

Answer 1

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Answer 2

To implicitly differentiate the given equation (x^2 - (x - 1)(x - y)^2 = \frac{x^2}{y^2}), follow these steps:

  1. Differentiate both sides of the equation with respect to (x).
  2. Use the product rule and chain rule where necessary.
  3. Solve for (\frac{dy}{dx}).

Let's go through the steps:

  1. Differentiate both sides of the equation with respect to (x):

[ \frac{d}{dx}[x^2 - (x - 1)(x - y)^2] = \frac{d}{dx}\left(\frac{x^2}{y^2}\right) ]

  1. Apply the differentiation rules:

On the left side: [ \frac{d}{dx}[x^2 - (x - 1)(x - y)^2] = 2x - \frac{d}{dx}[(x - 1)(x - y)^2] ]

On the right side: [ \frac{d}{dx}\left(\frac{x^2}{y^2}\right) = \frac{d}{dx}(x^2)y^{-2} + x^2\frac{d}{dx}(y^{-2}) ]

  1. Compute the derivatives:

For the left side: [ \frac{d}{dx}[(x - 1)(x - y)^2] = (x - 1)\frac{d}{dx}[(x - y)^2] + (x - y)^2\frac{d}{dx}(x - 1) ]

For the right side: [ \frac{d}{dx}(x^2)y^{-2} = 2xy^{-2} ] [ \frac{d}{dx}(y^{-2}) = -2y^{-3}\frac{dy}{dx} ]

Now, substitute all the derivatives back into the original equation and solve for (\frac{dy}{dx}). After simplification, you should have an expression for (\frac{dy}{dx}).

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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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