# How do you find #(dy)/(dx)# given #cos(2y)=sqrt(1-x^2)#?

##### 1 Answer

Dec 13, 2016

# dy/dx = x/(2 sin(2y)sqrt(1 - x^2)) #

or equivalently:

# dy/dx = 1/(2sqrt(1 - x^2)) #

#### Explanation:

Differentiating implicitly and applying the chain rule we get:

So we can rearrange to get;

We can also get an explicit expression should we need it;

Using

# sin^2 2y+cos^2 2y=1 #

# :. sin^2 2y+(1-x^2)=1 #

# :. sin^2 2y=x^2 #

# :. sin 2y=x #

So the earlier solution can be written as:

# \ \ \ \ \ dy/dx = x/(2xsqrt(1 - x^2)) #

# :. dy/dx = 1/(2sqrt(1 - x^2)) #