derivative of 9x^4y^8-2x^my^5-x^9y^n

Lösung

\frac{d}{d x} (9 x^{4} y^{8} - 2 x^{m} y^{5} - x^{9} y^{n})

36 x^{3} y^{8} + 72 x^{4} y^{7} \frac{d y}{d x} - 2 m x^{m - 1} y^{5} - 10 x^{m} y^{4} \frac{d y}{d x} - 9 x^{8} y^{n} - n x^{9} \frac{d y}{d x} y^{n - 1}

Schritte zur Lösung

\frac{d}{d x} (9 x^{4} y^{8} - 2 x^{m} y^{5} - x^{9} y^{n})

Wende die Summen-/Differenzregel an:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (9 x^{4} y^{8}) - \frac{d}{d x} (2 x^{m} y^{5}) - \frac{d}{d x} (x^{9} y^{n})

\frac{d}{d x} (9 x^{4} y^{8}) = 9 (4 x^{3} y^{8} + 8 y^{7} \frac{d y}{d x} x^{4})

\frac{d}{d x} (2 x^{m} y^{5}) = 2 (m x^{m - 1} y^{5} + 5 y^{4} \frac{d y}{d x} x^{m})

\frac{d}{d x} (x^{9} y^{n}) = 9 x^{8} y^{n} + n y^{n - 1} \frac{d y}{d x} x^{9}

= 9 (4 x^{3} y^{8} + 8 y^{7} \frac{d y}{d x} x^{4}) - 2 (m x^{m - 1} y^{5} + 5 y^{4} \frac{d y}{d x} x^{m}) - (9 x^{8} y^{n} + n y^{n - 1} \frac{d y}{d x} x^{9})

Vereinfache

9 (4 x^{3} y^{8} + 8 y^{7} \frac{d y}{d x} x^{4}) - 2 (m x^{m - 1} y^{5} + 5 y^{4} \frac{d y}{d x} x^{m}) - (9 x^{8} y^{n} + n y^{n - 1} \frac{d y}{d x} x^{9}) : 36 x^{3} y^{8} + 72 x^{4} y^{7} \frac{d y}{d x} - 2 m x^{m - 1} y^{5} - 10 x^{m} y^{4} \frac{d y}{d x} - 9 x^{8} y^{n} - n x^{9} \frac{d y}{d x} y^{n - 1}

= 36 x^{3} y^{8} + 72 x^{4} y^{7} \frac{d y}{d x} - 2 m x^{m - 1} y^{5} - 10 x^{m} y^{4} \frac{d y}{d x} - 9 x^{8} y^{n} - n x^{9} \frac{d y}{d x} y^{n - 1}

\int_{0}^{1} e^{x^{5}} d x

n = 5 \sum \infty \frac{7 ^{n}}{1 0 ^{n}}

y^{^{'}} - y = 4 t e^{2 t}, y (0) = 1

t \cdot y^{^{'}} + t \cdot y = 1 - y

\frac{\partial}{\partial x} (x e^{4 x y})