What is the derivative of (ln(5x)/(sqrt(x))) ?

The derivative of (ln(5x)/(sqrt(x))) is (2-ln(5x))/(2xsqrt(x))

What is the first derivative of (ln(5x)/(sqrt(x))) ?

The first derivative of (ln(5x)/(sqrt(x))) is (2-ln(5x))/(2xsqrt(x))

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\frac{d}{d x} (\frac{ln ( 5 x )}{x})

\frac{2 - ln ( 5 x )}{2 x x}

Solution steps

\frac{d}{d x} (\frac{ln ( 5 x )}{x})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( ln ( 5 x ) ) x - \frac{d}{d x} ( x ) ln ( 5 x )}{( x ) ^{2}}

\frac{d}{d x} (ln (5 x)) = \frac{1}{x}

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

= \frac{\frac{1}{x} x - \frac{1}{2 x} ln ( 5 x )}{( x ) ^{2}}

Simplify

\frac{\frac{1}{x} x - \frac{1}{2 x} ln ( 5 x )}{( x ) ^{2}} : \frac{2 - ln ( 5 x )}{2 x x}

= \frac{2 - ln ( 5 x )}{2 x x}

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What is the derivative of (ln(5x)/(sqrt(x))) ?
The derivative of (ln(5x)/(sqrt(x))) is (2-ln(5x))/(2xsqrt(x))
What is the first derivative of (ln(5x)/(sqrt(x))) ?
The first derivative of (ln(5x)/(sqrt(x))) is (2-ln(5x))/(2xsqrt(x))