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

Problem

tan(x+y)=x

Solution

1. Differentiate both sides with respect to x using the chain rule for the left side.

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

2. Apply the chain rule to the tangent function, where the derivative of tan(u) is sec2(u)

sec2(x+y)⋅d(x+y)/d(x)=1

3. Differentiate the inner term (x+y) with respect to x

sec2(x+y)⋅(1+d(y)/d(x))=1

4. Distribute the sec2(x+y) term to isolate the derivative.

sec2(x+y)+sec2(x+y)d(y)/d(x)=1

5. Subtract sec2(x+y) from both sides.

sec2(x+y)d(y)/d(x)=1−sec2(x+y)

6. Solve for dy/dx by dividing both sides by sec2(x+y)

d(y)/d(x)=(1−sec2(x+y))/sec2(x+y)

7. Simplify the expression using the trigonometric identity 1−sec2(θ)=−tan2(θ)

d(y)/d(x)=(−tan2(x+y))/sec2(x+y)

8. Convert to sines and cosines to simplify further.

d(y)/d(x)=−sin2(x+y)/cos2(x+y)⋅cos2(x+y)

9. Final simplification results in the squared sine function.

d(y)/d(x)=−sin2(x+y)

Final Answer

d(y)/d(x)=−sin2(x+y)

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