derivative of ((1+sin(x)/(1-cos(x)))^2)

解答

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

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

求解步骤

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

使用连锁律:

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

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

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

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

化简

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

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

\int_{- 3}^{2} (- x^{2} - x + 6) d x

\int (3 y^{2} + 6 y) (y^{3} + 3 y^{2} + 1)^{\frac{2}{3}} d y

t an g e n t f (x) = x^{2} - 8

\frac{d}{d x} (\frac{x + 3}{∣ x + 3 ∣})

x \to 0 + lim (\frac{( ln ( x ) ) ^{2}}{x})