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Answer 1 · 2017-01-28T17:22:54

Use the properties of logarithms to break into two terms and the differentiate each term.

$\frac{d y}{d x} = \frac{1}{2 x + 18} - \frac{1}{2 x - 18}$ $dy/dx = 1/(2x + 18) - 1/(2x - 18)$

$y = ln \sqrt{\frac{x + 9}{x - 9}}$ $y = lnsqrt((x+9)/(x-9))$

Use $ln (\sqrt{a}) = \frac{1}{2} ln (a)$ $ln(sqrt(a)) = 1/2ln(a)$

$y = \frac{1}{2} ln (\frac{x + 9}{x - 9})$ $y = 1/2ln((x+9)/(x-9))$

Use $ln (\frac{a}{b}) = ln (a) - ln (b)$ $ln(a/b) = ln(a) - ln(b)$

$y = \frac{1}{2} ln (x + 9) - \frac{1}{2} ln (x - 9)$ $y = 1/2ln(x+9)- 1/2ln(x-9)$

$\frac{d y}{d x} = \frac{1}{2 (x + 9)} - \frac{1}{2 (x - 9)}$ $dy/dx = 1/(2(x+9))- 1/(2(x-9))$

$\frac{d y}{d x} = \frac{1}{2 x + 18} - \frac{1}{2 x - 18}$ $dy/dx = 1/(2x + 18) - 1/(2x - 18)$