derivative of (x^4+y^4/(xy))

Lösung

\frac{d}{d x} (\frac{x ^{4} + y ^{4}}{x y})

\frac{- x ^{5} \frac{d y}{d x} + 3 x ^{4} y + 3 x y ^{4} \frac{d y}{d x} - y ^{5}}{x ^{2} y ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{4} + y ^{4}}{x y})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x ^{4} + y ^{4} ) x y - \frac{d}{d x} ( x y ) ( x ^{4} + y ^{4} )}{( x y ) ^{2}}

\frac{d}{d x} (x^{4} + y^{4}) = 4 x^{3} + 4 y^{3} \frac{d y}{d x}

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

= \frac{( 4 x ^{3} + 4 y ^{3} \frac{d y}{d x} ) x y - ( y + x \frac{d y}{d x} ) ( x ^{4} + y ^{4} )}{( x y ) ^{2}}

Vereinfache

\frac{( 4 x ^{3} + 4 y ^{3} \frac{d y}{d x} ) x y - ( y + x \frac{d y}{d x} ) ( x ^{4} + y ^{4} )}{( x y ) ^{2}} : \frac{- x ^{5} \frac{d y}{d x} + 3 x ^{4} y + 3 x y ^{4} \frac{d y}{d x} - y ^{5}}{x ^{2} y ^{2}}

= \frac{- x ^{5} \frac{d y}{d x} + 3 x ^{4} y + 3 x y ^{4} \frac{d y}{d x} - y ^{5}}{x ^{2} y ^{2}}

t an g e n t f (x) = - 3 x^{2} - 2 x, at x = 2

n = 0 \sum \infty (\frac{4}{10})^{n}

\int_{2}^{3} 21 t^{2} d t

\frac{d}{d x} (4 x + 12 y, 12 x - 3 y)

f (r) = 6 3 r