What is the derivative of-(ln(1+6x)/x) ?

The derivative of-(ln(1+6x)/x) is -(6x-ln(1+6x)(1+6x))/(x^2(1+6x))

What is the first derivative of-(ln(1+6x)/x) ?

The first derivative of-(ln(1+6x)/x) is -(6x-ln(1+6x)(1+6x))/(x^2(1+6x))

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\frac{d}{d x} (- \frac{ln ( 1 + 6 x )}{x})

- \frac{6 x - ln ( 1 + 6 x ) ( 1 + 6 x )}{x ^{2} ( 1 + 6 x )}

Solution steps

\frac{d}{d x} (- \frac{ln ( 1 + 6 x )}{x})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= - \frac{d}{d x} (\frac{ln ( 1 + 6 x )}{x})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= - \frac{\frac{d}{d x} ( ln ( 1 + 6 x ) ) x - \frac{d x}{d x} ln ( 1 + 6 x )}{x ^{2}}

\frac{d}{d x} (ln (1 + 6 x)) = \frac{6}{1 + 6 x}

\frac{d x}{d x} = 1

= - \frac{\frac{6}{1 + 6 x} x - 1 \cdot ln ( 1 + 6 x )}{x ^{2}}

Simplify

- \frac{\frac{6}{1 + 6 x} x - 1 \cdot ln ( 1 + 6 x )}{x ^{2}} : - \frac{6 x - ln ( 1 + 6 x ) ( 1 + 6 x )}{x ^{2} ( 1 + 6 x )}

= - \frac{6 x - ln ( 1 + 6 x ) ( 1 + 6 x )}{x ^{2} ( 1 + 6 x )}

What is the derivative of-(ln(1+6x)/x) ?
The derivative of-(ln(1+6x)/x) is -(6x-ln(1+6x)(1+6x))/(x^2(1+6x))
What is the first derivative of-(ln(1+6x)/x) ?
The first derivative of-(ln(1+6x)/x) is -(6x-ln(1+6x)(1+6x))/(x^2(1+6x))