Find dy/dx y=sin(x+y)

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, noting that the derivative of the inner function x+y is 1+d(y)/d(x)

d(y)/d(x)=cos(x+y)⋅(1+d(y)/d(x))

3. Distribute the cosine term to separate the terms containing d(y)/d(x)

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

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

d(y)/d(x)−cos(x+y)d(y)/d(x)=cos(x+y)

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

d(y)/d(x)*(1−cos(x+y))=cos(x+y)

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

d(y)/d(x)=cos(x+y)/(1−cos(x+y))

Final Answer

d(y)/d(x)=cos(x+y)/(1−cos(x+y))

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