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Differentiate with respect to x the implicit function sin(y)+x^2y^3-cos(x)=2y​

User Nirma
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Answer:

To differentiate the implicit function sin(y) + x^2y^3 - cos(x) = 2y with respect to x, we will use the chain rule and product rule.

First, we will take the derivative of both sides of the equation with respect to x:

d/dx [sin(y) + x^2y^3 - cos(x)] = d/dx [2y]

Next, we will differentiate each term on the left side of the equation:

cos(x) - 2xy^2 dx/dx + 3x^2y^2 + sin(y) dy/dx = 2 dy/dx

We can simplify this equation by moving all the terms involving dy/dx to the left side:

cos(x) - 2xy^2 - 2 dy/dx sin(y) = dy/dx [2 - sin(y)]

Now we can solve for dy/dx by isolating it on one side of the equation:

dy/dx [2 - sin(y)] = cos(x) - 2xy^2

dy/dx = (cos(x) - 2xy^2) / [2 - sin(y)]

Therefore, the derivative of the implicit function sin(y) + x^2y^3 - cos(x) = 2y with respect to x is:

dy/dx = (cos(x) - 2xy^2) / [2 - sin(y)]

User Alex Correia
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