What is the derivative of ln(sqrt(((x-4)/(x-3)))) ?

The derivative of ln(sqrt(((x-4)/(x-3)))) is 1/(2(x-3)(x-4))

What is the first derivative of ln(sqrt(((x-4)/(x-3)))) ?

The first derivative of ln(sqrt(((x-4)/(x-3)))) is 1/(2(x-3)(x-4))

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\frac{d}{d x} (ln (\frac{( x - 4 )}{x - 3}))

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

Solution steps

\frac{d}{d x} (ln (\frac{( x - 4 )}{x - 3}))

Apply the chain rule:

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

= \frac{1}{\frac{x - 4}{x - 3}} \frac{d}{d x} (\frac{x - 4}{x - 3})

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

= \frac{1}{\frac{x - 4}{x - 3}} \cdot \frac{1}{2 x - 4 ( x - 3 ) ^{\frac{3}{2}}}

Simplify

\frac{1}{\frac{x - 4}{x - 3}} \cdot \frac{1}{2 x - 4 ( x - 3 ) ^{\frac{3}{2}}} : \frac{1}{2 ( x - 3 ) ( x - 4 )}

= \frac{1}{2 ( x - 3 ) ( x - 4 )}

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x^{2} - 1, x + 1

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What is the derivative of ln(sqrt(((x-4)/(x-3)))) ?
The derivative of ln(sqrt(((x-4)/(x-3)))) is 1/(2(x-3)(x-4))
What is the first derivative of ln(sqrt(((x-4)/(x-3)))) ?
The first derivative of ln(sqrt(((x-4)/(x-3)))) is 1/(2(x-3)(x-4))