Differentiating equations of the type ln[f(x)]

To solve such equations we take advantage of log lawes to simplify the problem .

E.g

ln[sqrt(1-x²)] = ln[(1-x²)^1/2] = 1/2ln[1-x²]

After simplifing the problem we can differentiate with respect to x 

y = 1/2ln[1-x²]

 let f(x) = 1-x²

Use the Chain rule 

dy/dx = dy/df * df/dx 

dy/df = 1/(2*f(x))

df/dx = -2x

dy/dx = - 1/2[  2x/( 1-x²  ) ]

Provides a good practice of chain rule. differentiating logarithms and properties of logs.