Find dy/dx 4cos(x)sin(y)=1

Problem

4*cos(x)*sin(y)=1

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the product rule to the left side.

(d(4)*cos(x)*sin(y))/d(x)=d(1)/d(x)

2. Apply the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) to the term 4*cos(x)*sin(y)

4*cos(x)d(sin(y))/d(x)+sin(y)(d(4)*cos(x))/d(x)=0

3. Apply the chain rule to differentiate sin(y) and the standard derivative for cos(x)

4*cos(x)*cos(y)d(y)/d(x)+sin(y)*(−4*sin(x))=0

4. Isolate the term containing d(y)/d(x) by moving the other term to the right side of the equation.

4*cos(x)*cos(y)d(y)/d(x)=4*sin(x)*sin(y)

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

d(y)/d(x)=(4*sin(x)*sin(y))/(4*cos(x)*cos(y))

6. Simplify the expression using the trigonometric identity tan(θ)=sin(θ)/cos(θ)

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

Final Answer

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

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