Find dy/dx cos(xy^2)=y

Problem

cos(x*y2)=y

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule.

d(cos(x*y2))/d(x)=d(y)/d(x)

2. Apply the chain rule to the left side, which requires the derivative of the outer function cos(u) and the inner function x*y2

−sin(x*y2)⋅(d(x)*y2)/d(x)=d(y)/d(x)

3. Apply the product rule to differentiate x*y2 where the derivative of y2 is 2*yd(y)/d(x)

−sin(x*y2)⋅(y2+x⋅2*yd(y)/d(x))=d(y)/d(x)

4. Distribute the −sin(x*y2) term to isolate the terms containing d(y)/d(x)

−y2*sin(x*y2)−2*x*y*sin(x*y2)d(y)/d(x)=d(y)/d(x)

5. Group all terms involving d(y)/d(x) on one side of the equation.

−y2*sin(x*y2)=d(y)/d(x)+2*x*y*sin(x*y2)d(y)/d(x)

6. Factor out d(y)/d(x) from the right side.

−y2*sin(x*y2)=d(y)/d(x)*(1+2*x*y*sin(x*y2))

7. Solve for dy/dx by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(−y2*sin(x*y2))/(1+2*x*y*sin(x*y2))

Final Answer

d(y)/d(x)=−(y2*sin(x*y2))/(1+2*x*y*sin(x*y2))

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