Find dy/dx y=sin(xy)

Problem

y=sin(x*y)

Solution

1. Differentiate both sides with respect to x using the chain rule on the right side.

d(y)/d(x)=d(sin(x*y))/d(x)

2. Apply the chain rule to the sine function, which requires taking the derivative of the inner function x*y

d(y)/d(x)=cos(x*y)⋅(d(x)*y)/d(x)

3. Apply the product rule to the inner derivative (d(x)*y)/d(x) where the derivative of x is 1 and the derivative of y is d(y)/d(x)

d(y)/d(x)=cos(x*y)⋅(y+xd(y)/d(x))

4. Distribute the cos(x*y) term into the parentheses.

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

5. Group the terms containing d(y)/d(x) on the left side of the equation.

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

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

d(y)/d(x)*(1−x*cos(x*y))=y*cos(x*y)

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

d(y)/d(x)=(y*cos(x*y))/(1−x*cos(x*y))

Final Answer

d(y)/d(x)=(y*cos(x*y))/(1−x*cos(x*y))

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