What is the derivative of (ln(4x)/(ln(2x))) ?

The derivative of (ln(4x)/(ln(2x))) is -(ln(2))/(xln^2(2x))

What is the first derivative of (ln(4x)/(ln(2x))) ?

The first derivative of (ln(4x)/(ln(2x))) is -(ln(2))/(xln^2(2x))

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\frac{d}{d x} (\frac{ln ( 4 x )}{ln ( 2 x )})

- \frac{ln ( 2 )}{x ln ^{2} ( 2 x )}

Solution steps

\frac{d}{d x} (\frac{ln ( 4 x )}{ln ( 2 x )})

Apply the Quotient Rule:

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

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

\frac{d}{d x} (ln (4 x)) = \frac{1}{x}

\frac{d}{d x} (ln (2 x)) = \frac{1}{x}

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

Simplify

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

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

What is the derivative of (ln(4x)/(ln(2x))) ?
The derivative of (ln(4x)/(ln(2x))) is -(ln(2))/(xln^2(2x))
What is the first derivative of (ln(4x)/(ln(2x))) ?
The first derivative of (ln(4x)/(ln(2x))) is -(ln(2))/(xln^2(2x))