Question 6.11.2: Find the domains of the following functions and compute thei......

(a) ln(1 − x) is defined if 1 − x > 0, that is if x < 1. To find its derivative, we use (6.11.2), with h(x) = 1 − x. Then h^{\prime}(x) = −1, and

y=\ln h(x)\Rightarrow y^{\prime}={\frac{h^{\prime}(x)}{h(x)}}     (6.11.2)

(b)   ln(4 − x²) is defined if 4 − x² > 0, that is if (2 − x)(2 + x) > 0. This is satisfied if and only if −2 < x < 2. Formula (6.11.2) gives

(c) We can write y = ln  u − \frac{1}{4}x, where u = (x − 1)/(x + 1). For the function to be defined, we require that u > 0. A sign diagram shows that this is satisfied if x < −1 or x > 1. Using (6.11.2), we obtain

where

So