derivative of 1/(sqrt(1+e^{4x)})

解答

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

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

求解步骤

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

使用指数法则:

\frac{1}{a} = a^{- 1}

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

使用连锁律:

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

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

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

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

化简

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

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

x \to 0 + lim (\frac{x + ∣ x ∣}{x})

x \to 4 lim (x^{3} + 6 x^{2} - 5 x - 1)

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

\int sec^{5} (x) d x

t an g e n t f (x) = x x, at x = 0