derivative of e^x(xsin(x+(1-x)cos(x)))

解答

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

2 e^{x} x sin (x)

求解步骤

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

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{x}, g = x sin (x) + (1 - x) cos (x)

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

\frac{d}{d x} (e^{x}) = e^{x}

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

= e^{x} (x sin (x) + (1 - x) cos (x)) + (x cos (x) + x sin (x) - cos (x)) e^{x}

化简

e^{x} (x sin (x) + (1 - x) cos (x)) + (x cos (x) + x sin (x) - cos (x)) e^{x} : 2 e^{x} x sin (x)

= 2 e^{x} x sin (x)

\frac{\partial}{\partial x} (\frac{1}{x ^{7}})

\int cos^{2} (5 y) sin (5 y) d y

\int_{- 3}^{\infty} e^{- x} d x

t a y l or \frac{1}{1 - x}

\frac{\partial}{\partial z} (- 9 z cos (3 x + yz))