Find dy/dx cos(xy)+x^3=y^3

Problem

cos(x*y)+x3=y3

Solution

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

d(cos(x*y))/d(x)+d(x3)/d(x)=d(y3)/d(x)

2. Apply the product rule to the term cos(x*y) which involves the derivative of the inner function x*y

−sin(x*y)⋅(y+xd(y)/d(x))+3*x2=3*y2d(y)/d(x)

3. Distribute the −sin(x*y) term to separate the terms containing d(y)/d(x)

−y*sin(x*y)−x*sin(x*y)d(y)/d(x)+3*x2=3*y2d(y)/d(x)

4. Group all terms containing d(y)/d(x) on one side of the equation and the remaining terms on the other.

3*x2−y*sin(x*y)=3*y2d(y)/d(x)+x*sin(x*y)d(y)/d(x)

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

3*x2−y*sin(x*y)=d(y)/d(x)*(3*y2+x*sin(x*y))

6. Solve for d(y)/d(x) by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(3*x2−y*sin(x*y))/(3*y2+x*sin(x*y))

Final Answer

d(y)/d(x)=(3*x2−y*sin(x*y))/(3*y2+x*sin(x*y))

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