What is the derivative of ln((5x^2+x+6/(x^3))) ?

The derivative of ln((5x^2+x+6/(x^3))) is (-5x^2-2x-18)/(x(5x^2+x+6))

What is the first derivative of ln((5x^2+x+6/(x^3))) ?

The first derivative of ln((5x^2+x+6/(x^3))) is (-5x^2-2x-18)/(x(5x^2+x+6))

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\frac{d}{d x} (ln (\frac{5 x ^{2} + x + 6}{x ^{3}}))

\frac{- 5 x ^{2} - 2 x - 18}{x ( 5 x ^{2} + x + 6 )}

Solution steps

\frac{d}{d x} (ln (\frac{5 x ^{2} + x + 6}{x ^{3}}))

Apply the chain rule:

\frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \frac{d}{d x} (\frac{5 x ^{2} + x + 6}{x ^{3}})

= \frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \frac{d}{d x} (\frac{5 x ^{2} + x + 6}{x ^{3}})

\frac{d}{d x} (\frac{5 x ^{2} + x + 6}{x ^{3}}) = \frac{- 5 x ^{2} - 2 x - 18}{x ^{4}}

= \frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \cdot \frac{- 5 x ^{2} - 2 x - 18}{x ^{4}}

Simplify

\frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \cdot \frac{- 5 x ^{2} - 2 x - 18}{x ^{4}} : \frac{- 5 x ^{2} - 2 x - 18}{x ( 5 x ^{2} + x + 6 )}

= \frac{- 5 x ^{2} - 2 x - 18}{x ( 5 x ^{2} + x + 6 )}

\int_{0}^{1} 12 e^{3 x} d x

\frac{s}{s ^{2} + 2 \cdot 3 \cdot π \cdot s + π ^{2}}

\frac{d}{d x} ((5 x + e^{x}) (5 - x))

\frac{d}{d t} (2 e^{2 t})

\int 4 x (x^{2} + 29000)^{- \frac{2}{3}} d x

What is the derivative of ln((5x^2+x+6/(x^3))) ?
The derivative of ln((5x^2+x+6/(x^3))) is (-5x^2-2x-18)/(x(5x^2+x+6))
What is the first derivative of ln((5x^2+x+6/(x^3))) ?
The first derivative of ln((5x^2+x+6/(x^3))) is (-5x^2-2x-18)/(x(5x^2+x+6))