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

Answer 1

#dy/dx = (y(2xsin(x^2 + 1)))/(cos(x^2) - x - 2y)#.

Firstly, let's apply the derivative operator to both sides, I'll try my best to colour code the use of the product rule:

#d/dx [color(red)(ycos(x^2)) color(teal)(- y^2)] = d/dx[color(purple)(x)color(orange)y]#
#color(red)(dy/dx * cos(x^2) + y * (-sin(x^2) * 2x)) color(teal)(- 2y * dy/dx) = color(purple)(1) * color(orange)(y) + color(purple)(x) * color(orange)(dy/dx)#
#dy/dx * cos(x^2) - 2y * dy/dx - x * dy/dx = 2xy * sin(x^2) + y #
#dy/dx (cos(x^2) - 2y - x) = 2xy * sin(x^2) + y#
#dy/dx = (2xy * sin(x^2) + y)/(cos(x^2) - 2y - x)#
#dy/dx = (y(2xsin(x^2 + 1)))/(cos(x^2) - x - 2y)#.
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Answer 2

To differentiate the equation ( y \cos(x^2) - y^2 = xy ) with respect to ( x ), you'll use the implicit differentiation technique.

First, differentiate each term of the equation with respect to ( x ), then solve for ( \frac{{dy}}{{dx}} ). The steps are as follows:

  1. Differentiate ( y \cos(x^2) ) using the product rule.
  2. Differentiate ( -y^2 ) with respect to ( x ).
  3. Differentiate ( xy ) using the product rule.
  4. Combine the results and solve for ( \frac{{dy}}{{dx}} ).

Applying these steps will give you the derivative of ( y ) with respect to ( x ), ( \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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