derivative y=(x/(1+x))^5

解答

导数

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

\frac{5 x ^{4}}{( 1 + x ) ^{6}}

求解步骤

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

使用连锁律:

5 (\frac{x}{1 + x})^{4} \frac{d}{d x} (\frac{x}{1 + x})

= 5 (\frac{x}{1 + x})^{4} \frac{d}{d x} (\frac{x}{1 + x})

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

= 5 (\frac{x}{1 + x})^{4} \frac{1}{( 1 + x ) ^{2}}

化简

5 (\frac{x}{1 + x})^{4} \frac{1}{( 1 + x ) ^{2}} : \frac{5 x ^{4}}{( 1 + x ) ^{6}}

= \frac{5 x ^{4}}{( 1 + x ) ^{6}}

\frac{d}{d x} (2 x \cdot e^{x^{2}})

x \to 25 lim (x - 5)

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

\int 2 + x - x^{2} d x

\int \frac{e ^{4 x}}{4 - e ^{4 x}} d x