derivative of ln((x^3+1/(x^2+1)))

解答

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

\frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{3} + 1 ) ( x ^{2} + 1 )}

求解步骤

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

使用连锁律:

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

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

\frac{d}{d x} (\frac{x ^{3} + 1}{x ^{2} + 1}) = \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{2} + 1 ) ^{2}}

= \frac{1}{\frac{x ^{3} + 1}{x ^{2} + 1}} \cdot \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{2} + 1 ) ^{2}}

化简

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

= \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{3} + 1 ) ( x ^{2} + 1 )}

\int 3 e^{3 t} d t

t an g e n t \frac{x}{x + 1}

\int \frac{20 x ^{3} - 40 x ^{2} + 3}{x ^{2} - 2 x} d x

d er i v a t i v e f (x) = \frac{x}{x - 6}

t an g e n t f (x) = 5 x^{2} + 7 x