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

Problem

tan(x/y)=x+y

Solution

1. Differentiate both sides with respect to x using the chain rule and the quotient rule for the term involving y

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

2. Apply the chain rule to the left side, noting that the derivative of tan(u) is sec2(u)

sec2(x/y)⋅d()/d(x)x/y=1+d(y)/d(x)

3. Apply the quotient rule to differentiate x/y

sec2(x/y)⋅(y(1)−xd(y)/d(x))/(y2)=1+d(y)/d(x)

4. Distribute the terms to isolate the components containing d(y)/d(x)

(y*sec2(x/y))/(y2)−(x*sec2(x/y)d(y)/d(x))/(y2)=1+d(y)/d(x)

5. Simplify the first fraction and move all terms with d(y)/d(x) to one side.

sec2(x/y)/y−1=d(y)/d(x)+(x*sec2(x/y))/(y2)d(y)/d(x)

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

(sec2(x/y)−y)/y=d(y)/d(x)*(1+(x*sec2(x/y))/(y2))

7. Find a common denominator for the expression inside the parentheses.

(sec2(x/y)−y)/y=d(y)/d(x)*((y2+x*sec2(x/y))/(y2))

8. Solve for dy/dx by multiplying both sides by the reciprocal of the term in parentheses.

d(y)/d(x)=(y2*(sec2(x/y)−y))/(y*(y2+x*sec2(x/y)))

9. Simplify the expression by canceling one factor of y

d(y)/d(x)=(y*(sec2(x/y)−y))/(y2+x*sec2(x/y))

Final Answer

d(y)/d(x)=(y*sec2(x/y)−y2)/(y2+x*sec2(x/y))

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