Find dy/dx sin(x)=x(1+tan(y))

Problem

sin(x)=x*(1+tan(y))

Solution

1. Differentiate both sides with respect to x applying the product rule to the right side and the chain rule to the term involving y

d(sin(x))/d(x)=(d(x)*(1+tan(y)))/d(x)

2. Apply the derivative rules where the derivative of sin(x) is cos(x) and the derivative of x*(1+tan(y)) follows the rule u′*v+u*v′

cos(x)=(1)*(1+tan(y))+x*(sec2(y)d(y)/d(x))

3. Distribute and isolate the term containing d(y)/d(x) by subtracting (1+tan(y)) from both sides of the equation.

cos(x)−1−tan(y)=x*sec2(y)d(y)/d(x)

4. Solve for dy/dx by dividing both sides by x*sec2(y)

d(y)/d(x)=(cos(x)−1−tan(y))/(x*sec2(y))

Final Answer

d(y)/d(x)=(cos(x)−1−tan(y))/(x*sec2(y))

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