x cos y = 2

Question

Find dy/dx by implicit differentiation

Answer 1

Hi, Ehsan.

 

Using the product formula, taking the derivative of: x cos y = 2,

 

We get:  1 cos y  + x (-sin y) dy/dx = 0

 

So -x sin y (dy/dx) = -cos y,   Divide by -xsiny, dy/dx = (cos y)/(x sin y) = (cot y)/x.

 

Since x = 2/(cos y), you may want to get the answer in terms of y only.

 

dy/dx = (cot y)(cos y)/2 = ½(cot y)(cos y).

Answer 2

Differentiate both sides of equation using the chain rule.

 

 

(xcos y)' = (2)'

 

cos y - xy'sin y = 0

 

- xy'sin y = -cos y

 

y' (- xsin y) = -cos y

 

y' = (cos y) / (xsin y)