f ( x , y ) = { x y 3 x 2 + y 2 if ( x , y ) ≠ ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) f(x,y) = \begin{cases}
\dfrac{xy^3 }{x^2 +y^2} &\text{if } (x, y)\not=(0, 0) \\
0 &\text{if } (x,y)=(0,0)
\end{cases} f ( x , y ) = ⎩ ⎨ ⎧ x 2 + y 2 x y 3 0 if ( x , y ) = ( 0 , 0 ) if ( x , y ) = ( 0 , 0 ) If ( x , y ) ≠ ( 0 , 0 ) (x, y)\not=(0, 0) ( x , y ) = ( 0 , 0 )
f x = ∂ f ∂ x = ∂ ∂ x ( x y 3 x 2 + y 2 ) = f_{x}=\dfrac{\partial f}{\partial x}=\dfrac{\partial }{\partial x}( \dfrac{xy^3 }{x^2 +y^2} )= f x = ∂ x ∂ f = ∂ x ∂ ( x 2 + y 2 x y 3 ) =
= y 3 ( x 2 + y 2 ) − 2 x 2 y 3 ( x 2 + y 2 ) 2 = y 3 ( y 2 − x 2 ) ( x 2 + y 2 ) 2 =\dfrac{y^3(x^2+y^2)-2x^2y^3 }{(x^2 +y^2)^2}=\dfrac{y^3(y^2-x^2)}{(x^2 +y^2)^2} = ( x 2 + y 2 ) 2 y 3 ( x 2 + y 2 ) − 2 x 2 y 3 = ( x 2 + y 2 ) 2 y 3 ( y 2 − x 2 )
f x ( 0 , 0 ) = lim h → 0 f ( 0 + h , 0 ) − f ( 0 , 0 ) h = f_x(0,0)=\lim\limits_{h\to 0}\dfrac{f(0+h,0)-f(0,0) }{h}= f x ( 0 , 0 ) = h → 0 lim h f ( 0 + h , 0 ) − f ( 0 , 0 ) =
= lim h → 0 f ( 0 + h , 0 ) − f ( 0 , 0 ) h = =\lim\limits_{h\to 0}\dfrac{f(0+h,0)-f(0,0) }{h}= = h → 0 lim h f ( 0 + h , 0 ) − f ( 0 , 0 ) =
= lim h → 0 h ( 0 ) 3 h 2 + ( 0 ) 2 − 0 h = 0 =\lim\limits_{h\to 0}\dfrac{ \dfrac{h(0)^3 }{h^2 +(0)^2}-0 }{h}=0 = h → 0 lim h h 2 + ( 0 ) 2 h ( 0 ) 3 − 0 = 0
f x ( x , y ) = { y 3 ( x 2 − y 2 ) ( x 2 + y 2 ) 2 if ( x , y ) ≠ ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) f_x(x,y) = \begin{cases}
\dfrac{y^3(x^2-y^2) }{(x^2 +y^2)^2} &\text{if } (x, y)\not=(0, 0) \\
0 &\text{if } (x,y)=(0,0)
\end{cases} f x ( x , y ) = ⎩ ⎨ ⎧ ( x 2 + y 2 ) 2 y 3 ( x 2 − y 2 ) 0 if ( x , y ) = ( 0 , 0 ) if ( x , y ) = ( 0 , 0 )
Let x = r c o s θ , y = r sin θ x=rcos\theta, y=r\sin\theta x = rcos θ , y = r sin θ
lim ( x , y ) → 0 f x ( x , y ) = lim ( x , y ) → 0 y 3 ( x 2 − y 2 ) ( x 2 + y 2 ) 2 = \lim\limits_{(x, y)\to 0}f_x(x, y)=\lim\limits_{(x, y)\to 0}\dfrac{y^3(x^2-y^2) }{(x^2 +y^2)^2}= ( x , y ) → 0 lim f x ( x , y ) = ( x , y ) → 0 lim ( x 2 + y 2 ) 2 y 3 ( x 2 − y 2 ) =
= lim r → 0 r 3 sin 3 θ r 2 ( cos 2 θ − sin 2 θ ) r 4 = =\lim\limits_{r\to 0}\dfrac{r^3\sin^3\theta r^2(\cos^2\theta-\sin^2\theta) }{r^4} = = r → 0 lim r 4 r 3 sin 3 θ r 2 ( cos 2 θ − sin 2 θ ) =
= lim r → 0 r sin 3 θ ( cos 2 θ − sin 2 θ ) = 0 = f x ( 0 , 0 ) =\lim\limits_{r\to 0}r\sin^3\theta(\cos^2\theta-\sin^2\theta)=0=f_x(0,0) = r → 0 lim r sin 3 θ ( cos 2 θ − sin 2 θ ) = 0 = f x ( 0 , 0 ) The function f x ( x , y ) f_x(x, y) f x ( x , y ) ( 0 , 0 ) . (0,0). ( 0 , 0 ) .
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