erweitern 6(x^2+y^2)^5(2x+2y(dy)/(dx))

Lösung

erweitern

6 (x^{2} + y^{2})^{5} (2 x + 2 y \frac{d y}{d x})

12 x^{11} + 60 x^{9} y^{2} + 120 x^{7} y^{4} + 120 x^{5} y^{6} + 60 x^{3} y^{8} + 12 x y^{10} + 12 x^{10} y \frac{d y}{d x} + 60 x^{8} y^{3} \frac{d y}{d x} + 120 x^{6} y^{5} \frac{d y}{d x} + 120 x^{4} y^{7} \frac{d y}{d x} + 60 x^{2} y^{9} \frac{d y}{d x} + 12 y^{11} \frac{d y}{d x}

Schritte zur Lösung

6 (x^{2} + y^{2})^{5} (2 x + 2 y \frac{d y}{d x})

Wende das Distributivgesetz an:

a (b + c) = ab + a c

a = 6 (x^{2} + y^{2})^{5}, b = 2 x, c = 2 y \frac{d y}{d x}

= 6 (x^{2} + y^{2})^{5} \cdot 2 x + 6 (x^{2} + y^{2})^{5} \cdot 2 y \frac{d y}{d x}

= 6 \cdot 2 x (x^{2} + y^{2})^{5} + 6 \cdot 2 y (x^{2} + y^{2})^{5} \frac{d y}{d x}

Vereinfache

6 \cdot 2 x (x^{2} + y^{2})^{5} + 6 \cdot 2 y (x^{2} + y^{2})^{5} \frac{d y}{d x} : 12 x^{11} + 60 x^{9} y^{2} + 120 x^{7} y^{4} + 120 x^{5} y^{6} + 60 x^{3} y^{8} + 12 x y^{10} + 12 x^{10} y \frac{d y}{d x} + 60 x^{8} y^{3} \frac{d y}{d x} + 120 x^{6} y^{5} \frac{d y}{d x} + 120 x^{4} y^{7} \frac{d y}{d x} + 60 x^{2} y^{9} \frac{d y}{d x} + 12 y^{11} \frac{d y}{d x}

= 12 x^{11} + 60 x^{9} y^{2} + 120 x^{7} y^{4} + 120 x^{5} y^{6} + 60 x^{3} y^{8} + 12 x y^{10} + 12 x^{10} y \frac{d y}{d x} + 60 x^{8} y^{3} \frac{d y}{d x} + 120 x^{6} y^{5} \frac{d y}{d x} + 120 x^{4} y^{7} \frac{d y}{d x} + 60 x^{2} y^{9} \frac{d y}{d x} + 12 y^{11} \frac{d y}{d x}

e x p an d 8 (2 a + \frac{aπ}{2})

e x p an d \frac{( 1 - e ^{- 3 π s} )}{s ( s ^{2} + 1 )} + \frac{1}{s ^{2} + 1}

e x p an d 4 (1 - sin^{2} (x))

e x p an d \frac{4 ( - 3 x ^{2} + 1 )}{( x ^{2} + 1 ) ^{3}}

e x p an d \frac{( 5 x - 3 )}{( ( x + 1 ) ( x - 3 ) )}