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

The derivative of ln((1+sqrt(x)/(x^3))) is (x^{5/2}-6x^2(sqrt(x)+1))/(2x^3(sqrt(x)+1))

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

The first derivative of ln((1+sqrt(x)/(x^3))) is (x^{5/2}-6x^2(sqrt(x)+1))/(2x^3(sqrt(x)+1))

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int_{\frac{1}{5}}^{7} 8 x ln (5 x) d x

y^{^{'}} + \frac{2}{x} y = 4 x

\frac{d}{d x} (sin^{2} (\frac{x}{2}))

\frac{d}{d x} (ln (7 x^{2} + 5 y^{2}))

4 t^{\frac{- 1}{8}} + 5 ln (t) + ln (5)

What is the derivative of ln((1+sqrt(x)/(x^3))) ?
The derivative of ln((1+sqrt(x)/(x^3))) is (x^{5/2}-6x^2(sqrt(x)+1))/(2x^3(sqrt(x)+1))
What is the first derivative of ln((1+sqrt(x)/(x^3))) ?
The first derivative of ln((1+sqrt(x)/(x^3))) is (x^{5/2}-6x^2(sqrt(x)+1))/(2x^3(sqrt(x)+1))