What is the derivative of ln((x^2/5+9/x)) ?

The derivative of ln((x^2/5+9/x)) is (2x^3-45)/(x(x^3+45))

What is the first derivative of ln((x^2/5+9/x)) ?

The first derivative of ln((x^2/5+9/x)) is (2x^3-45)/(x(x^3+45))

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\frac{d}{d x} (ln (\frac{x ^{2}}{5} + \frac{9}{x}))

\frac{2 x ^{3} - 45}{x ( x ^{3} + 45 )}

Solution steps

\frac{d}{d x} (ln (\frac{x ^{2}}{5} + \frac{9}{x}))

Apply the chain rule:

\frac{1}{\frac{x ^{2}}{5} + \frac{9}{x}} \frac{d}{d x} (\frac{x ^{2}}{5} + \frac{9}{x})

= \frac{1}{\frac{x ^{2}}{5} + \frac{9}{x}} \frac{d}{d x} (\frac{x ^{2}}{5} + \frac{9}{x})

\frac{d}{d x} (\frac{x ^{2}}{5} + \frac{9}{x}) = \frac{2 x}{5} - \frac{9}{x ^{2}}

= \frac{1}{\frac{x ^{2}}{5} + \frac{9}{x}} (\frac{2 x}{5} - \frac{9}{x ^{2}})

Simplify

\frac{1}{\frac{x ^{2}}{5} + \frac{9}{x}} (\frac{2 x}{5} - \frac{9}{x ^{2}}) : \frac{2 x ^{3} - 45}{x ( x ^{3} + 45 )}

= \frac{2 x ^{3} - 45}{x ( x ^{3} + 45 )}

y = 2 x^{2} - x, (1, 1)

\frac{d}{d x} (5 x + \frac{7}{x})

\int_{1}^{5} \frac{ln ( x )}{x ^{5}} d x

f (x) = 4 x^{3} - 11 x^{2} - 14 x + 19

x \to 0 lim ((\frac{1 + tan ( x )}{1 + sin ( x )})^{\frac{1}{s i n ( x )}})

What is the derivative of ln((x^2/5+9/x)) ?
The derivative of ln((x^2/5+9/x)) is (2x^3-45)/(x(x^3+45))
What is the first derivative of ln((x^2/5+9/x)) ?
The first derivative of ln((x^2/5+9/x)) is (2x^3-45)/(x(x^3+45))