Find dy/dx tan(x-y)=y/(1+x^2)

Problem

tan(x−y)=y/(1+x^2)

Solution

1. Differentiate both sides with respect to x using the chain rule on the left and the quotient rule on the right.

d(tan(x−y))/d(x)=d()/d(x)y/(1+x^2)

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

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

3. Distribute the sec^2(x−y) term on the left side to isolate the derivative terms.

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

4. Clear the fraction by multiplying both sides of the equation by (1+x^2)^2

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

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

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

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

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

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

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

8. Simplify the expression by factoring out (1+x^2) from the denominator.

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

Final Answer

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

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