(\partial)/(\partial x)(8xcos(5xy))

Lösung

\frac{\partial}{\partial x} (8 x cos (5 x y))

8 (cos (5 x y) - 5 x y sin (5 x y))

Schritte zur Lösung

\frac{\partial}{\partial x} (8 x cos (5 x y))

Behandle

y

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= 8 \frac{\partial}{\partial x} (x cos (5 x y))

Wende die Produktregel an:

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

f = x, g = cos (5 x y)

= 8 (\frac{\partial}{\partial x} (x) cos (5 x y) + \frac{\partial}{\partial x} (cos (5 x y)) x)

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (cos (5 x y)) = - sin (5 x y) \cdot 5 y

= 8 (1 \cdot cos (5 x y) + (- sin (5 x y) \cdot 5 y) x)

Vereinfache

= 8 (cos (5 x y) - 5 x y sin (5 x y))

x \to 0 + lim (\frac{0}{x})

\frac{d y}{d x} = \frac{1}{y + x}

t an g e n t f (x) = x^{2} + 7, at x = 3

\frac{\partial}{\partial y} (- 3 x e^{- x y})

\int e^{- 2 x} sin (e^{- x}) d x