Find dy/dx xcos(y)=1

Problem

x*cos(y)=1

Solution

1. Differentiate both sides with respect to x using implicit differentiation.

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

2. Apply the product rule to the left side, where the product rule is (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

xd(cos(y))/d(x)+cos(y)d(x)/d(x)=0

3. Apply the chain rule to the term involving y noting that d(cos(y))/d(x)=−sin(y)d(y)/d(x)

x*(−sin(y)d(y)/d(x))+cos(y)*(1)=0

4. Simplify the expression to prepare for solving for d(y)/d(x)

−x*sin(y)d(y)/d(x)+cos(y)=0

5. Isolate the term containing d(y)/d(x) by subtracting cos(y) from both sides.

−x*sin(y)d(y)/d(x)=−cos(y)

6. Solve for dy/dx by dividing both sides by −x*sin(y)

d(y)/d(x)=(−cos(y))/(−x*sin(y))

7. Simplify the fraction using the trigonometric identity cos(y)/sin(y)=cot(y)

d(y)/d(x)=cot(y)/x

Final Answer

d(y)/d(x)=cot(y)/x

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