(\partial)/(\partial x)(x^2(1+sin(1/x)))

解答

\frac{\partial}{\partial x} (x^{2} (1 + sin (\frac{1}{x})))

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

求解步骤

\frac{\partial}{\partial x} (x^{2} (1 + sin (\frac{1}{x})))

使用微分乘法定则:

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

f = x^{2}, g = 1 + sin (\frac{1}{x})

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

化简

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

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

x \to 2.5 lim (g (x))

\int \frac{1}{t ^{2} + 6 t + 25} d t

s l o p e (3, 2), (7, - 4)

a re a x^{2}, 2 x - x^{2}, 0, 1

\int x (x^{4} - 1)^{2} d x