Find dy/dx tan(x-y)=y/(8+x^2)

Problem

tan(x−y)=y/(8+x2)

Solution

1. Rewrite the equation to clear the fraction by multiplying both sides by 8+x2

(8+x2)*tan(x−y)=y

2. Differentiate both sides with respect to x using the product rule and the chain rule.

(d(8+x2)*tan(x−y))/d(x)=d(y)/d(x)

3. Apply the product rule to the left side.

(2*x)*tan(x−y)+(8+x2)*sec2(x−y)d(x−y)/d(x)=d(y)/d(x)

4. Differentiate the inner term (x−y) with respect to x

2*x*tan(x−y)+(8+x2)*sec2(x−y)*(1−d(y)/d(x))=d(y)/d(x)

5. Distribute the term (8+x2)*sec2(x−y) into the parentheses.

2*x*tan(x−y)+(8+x2)*sec2(x−y)−(8+x2)*sec2(x−y)d(y)/d(x)=d(y)/d(x)

6. Group all terms containing d(y)/d(x) on one side of the equation.

2*x*tan(x−y)+(8+x2)*sec2(x−y)=d(y)/d(x)+(8+x2)*sec2(x−y)d(y)/d(x)

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

2*x*tan(x−y)+(8+x2)*sec2(x−y)=d(y)/d(x)*(1+(8+x2)*sec2(x−y))

8. Solve for d(y)/d(x) by dividing both sides by the factor (1+(8+x2)*sec2(x−y))

d(y)/d(x)=(2*x*tan(x−y)+(8+x2)*sec2(x−y))/(1+(8+x2)*sec2(x−y))

Final Answer

d(y)/d(x)=(2*x*tan(x−y)+(8+x2)*sec2(x−y))/(1+(8+x2)*sec2(x−y))

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