What is the derivative of ln(sqrt((x^3/(x^5+1)))) ?

The derivative of ln(sqrt((x^3/(x^5+1)))) is (-2x^5+3)/(2x(x^5+1))

What is the first derivative of ln(sqrt((x^3/(x^5+1)))) ?

The first derivative of ln(sqrt((x^3/(x^5+1)))) is (-2x^5+3)/(2x(x^5+1))

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

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

Solution steps

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

Apply the chain rule:

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

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

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

= \frac{1}{\frac{x ^{3}}{x ^{5} + 1}} \cdot \frac{x ^{\frac{3}{2}} ( - 2 x ^{5} + 3 )}{2 x ^{2} ( x ^{5} + 1 ) ^{\frac{3}{2}}}

Simplify

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

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

y = 10 + tan^{2} (x)

\int \frac{cot ( β )}{sin ^{6} ( β )} d β

\int \frac{1}{1 + \frac{1}{x}} d x

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

\frac{d}{d x} (\frac{8}{3 + x ^{2}})

What is the derivative of ln(sqrt((x^3/(x^5+1)))) ?
The derivative of ln(sqrt((x^3/(x^5+1)))) is (-2x^5+3)/(2x(x^5+1))
What is the first derivative of ln(sqrt((x^3/(x^5+1)))) ?
The first derivative of ln(sqrt((x^3/(x^5+1)))) is (-2x^5+3)/(2x(x^5+1))