derivative of ((1-cos(x)/(1+cos(x)))^3)

解答

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

\frac{6 sin ( x ) ( 1 - cos ( x ) ) ^{2}}{( 1 + cos ( x ) ) ^{4}}

求解步骤

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

使用连锁律:

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

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

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

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

化简

3 (\frac{1 - cos ( x )}{1 + cos ( x )})^{2} \frac{2 sin ( x )}{( 1 + cos ( x ) ) ^{2}} : \frac{6 sin ( x ) ( 1 - cos ( x ) ) ^{2}}{( 1 + cos ( x ) ) ^{4}}

= \frac{6 sin ( x ) ( 1 - cos ( x ) ) ^{2}}{( 1 + cos ( x ) ) ^{4}}

x \to 0 lim (\frac{5 - 5 sin ( x )}{x})

x \to - \frac{π}{2} - lim (sec (x))

\frac{d}{d x} (tanh (x^{2} - 9))

x \to 7 lim (x^{4})

n = 1 \sum \infty \frac{( ( - 1 ) ^{(n - 1)} )}{( ( 2 n - 1 ) ^{3} )}