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)) #