# How do you calculate #y^2x = 2sin(xy^3) #For implicit differentiation?

The answer is

Let

Then,

Therefore,

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To calculate the derivative of ( y^2x = 2\sin(xy^3) ) using implicit differentiation, follow these steps:

- Differentiate both sides of the equation with respect to ( x ).
- Apply the product rule and chain rule as necessary.
- Solve for ( \frac{{dy}}{{dx}} ) to find the derivative.

Let's differentiate both sides of the equation:

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

Apply the product rule and chain rule:

[ y^2\frac{{d}}{{dx}}(x) + x\frac{{d}}{{dx}}(y^2) = 2\frac{{d}}{{dx}}(\sin(xy^3)) ]

[ y^2 + x\frac{{d}}{{dx}}(y^2) = 2\cos(xy^3)\frac{{d}}{{dx}}(xy^3) ]

[ y^2 + x\frac{{d}}{{dx}}(y^2) = 2\cos(xy^3)(y^3\frac{{d}}{{dx}}(x) + x\frac{{d}}{{dx}}(y^3)) ]

[ y^2 + x\frac{{d}}{{dx}}(y^2) = 2\cos(xy^3)(y^3 + x\frac{{d}}{{dx}}(y^3)) ]

Now, differentiate ( y^2 ) with respect to ( x ):

[ \frac{{d}}{{dx}}(y^2) = 2y\frac{{dy}}{{dx}} ]

Substitute this back into the equation:

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

Now, solve for ( \frac{{dy}}{{dx}} ):

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

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

[ \frac{{dy}}{{dx}}(2xy - 2x\cos(xy^3)\frac{{dy}}{{dx}}) = y^2(2y\cos(xy^3) - 1) ]

[ \frac{{dy}}{{dx}} = \frac{{y^2(2y\cos(xy^3) - 1)}}{{2xy - 2x\cos(xy^3)\frac{{dy}}{{dx}}}} ]

[ \frac{{dy}}{{dx}}(1 + 2x^2\cos(xy^3)) = y^2(2y\cos(xy^3) - 1) ]

[ \frac{{dy}}{{dx}} = \frac{{y^2(2y\cos(xy^3) - 1)}}{{1 + 2x^2\cos(xy^3)}} ]

So, the derivative of ( y^2x = 2\sin(xy^3) ) with respect to ( x ) using implicit differentiation is:

[ \frac{{dy}}{{dx}} = \frac{{y^2(2y\cos(xy^3) - 1)}}{{1 + 2x^2\cos(xy^3)}} ]

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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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