What is the derivative of ln(sqrt((a+bx/(a-bx)))) ?

The derivative of ln(sqrt((a+bx/(a-bx)))) is (ab)/((-bx+a)(bx+a))

What is the first derivative of ln(sqrt((a+bx/(a-bx)))) ?

The first derivative of ln(sqrt((a+bx/(a-bx)))) is (ab)/((-bx+a)(bx+a))

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\frac{d}{d x} (ln (\frac{a + b x}{a - b x}))

\frac{ab}{( - b x + a ) ( b x + a )}

Solution steps

\frac{d}{d x} (ln (\frac{a + b x}{a - b x}))

Apply the chain rule:

\frac{1}{\frac{a + b x}{a - b x}} \frac{d}{d x} (\frac{a + b x}{a - b x})

= \frac{1}{\frac{a + b x}{a - b x}} \frac{d}{d x} (\frac{a + b x}{a - b x})

\frac{d}{d x} (\frac{a + b x}{a - b x}) = \frac{ab}{b x + a ( - b x + a ) ^{\frac{3}{2}}}

= \frac{1}{\frac{a + b x}{a - b x}} \cdot \frac{ab}{b x + a ( - b x + a ) ^{\frac{3}{2}}}

Simplify

\frac{1}{\frac{a + b x}{a - b x}} \cdot \frac{ab}{b x + a ( - b x + a ) ^{\frac{3}{2}}} : \frac{ab}{( - b x + a ) ( b x + a )}

= \frac{ab}{( - b x + a ) ( b x + a )}

\frac{1}{s ^{2} + 4 s + 12}

\frac{d}{d x} ((7 x + 3)^{2})

\frac{d}{d x} ((3 x^{3} + 1) (- 4 x^{2} - 3)^{4})

x \to 4 lim (\frac{x ^{2} - 6}{x - 4})

\int \frac{t}{1 + t} d t

What is the derivative of ln(sqrt((a+bx/(a-bx)))) ?
The derivative of ln(sqrt((a+bx/(a-bx)))) is (ab)/((-bx+a)(bx+a))
What is the first derivative of ln(sqrt((a+bx/(a-bx)))) ?
The first derivative of ln(sqrt((a+bx/(a-bx)))) is (ab)/((-bx+a)(bx+a))