derivative of (x^2+y^2sin(1/(x^2+y^2)))

解答

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

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

求解步骤

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

使用微分乘法定则:

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

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

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

\frac{d}{d x} (x^{2} + y^{2}) = 2 x + 2 y \frac{d y}{d x}

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

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

化简

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

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

\int sin (x) in (cos (x)) d x

d er i v a t i v e f (x) = 8 sec (x) tan (x)

t an g e n t f (x) = x^{2} + 1, (0, 1)

n = 5 \sum \infty \frac{3 ^{n}}{1 2 ^{n}}

t an g e n t y = x^{2} - 5