derivative of sin(1/x*x^3)

解答

\frac{d}{d x} (sin (\frac{1}{x}) \cdot x^{3})

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

求解步骤

\frac{d}{d x} (sin (\frac{1}{x}) x^{3})

使用微分乘法定则:

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

f = sin (\frac{1}{x}), g = x^{3}

= \frac{d}{d x} (sin (\frac{1}{x})) x^{3} + \frac{d}{d x} (x^{3}) sin (\frac{1}{x})

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

\frac{d}{d x} (x^{3}) = 3 x^{2}

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

化简

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

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

d er i v a t i v e cos (sin (tan (5 x)))

x \to - 1 lim (x^{3} - 1)

\int \frac{cos ( \frac{π}{x ^{15}} )}{x ^{16}} d x

d er i v a t i v e 6 x e^{x}

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