Find dy/dx sin(xy)=x^2-y

Problem

sin(x*y)=x2−y

Solution

1. Differentiate both sides with respect to x treating y as a function of x (y(x).

d(sin(x*y))/d(x)=d(x2−y)/d(x)

2. Apply the chain rule to the left side and the product rule to the inner term x*y

cos(x*y)⋅(y+xd(y)/d(x))=2*x−d(y)/d(x)

3. Distribute the cos(x*y) term to isolate the terms containing d(y)/d(x)

y*cos(x*y)+x*cos(x*y)d(y)/d(x)=2*x−d(y)/d(x)

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

x*cos(x*y)d(y)/d(x)+d(y)/d(x)=2*x−y*cos(x*y)

5. Factor out d(y)/d(x) from the left side of the equation.

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

6. Solve for dy/dx by dividing both sides by the factor (x*cos(x*y)+1)

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

Final Answer

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

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