(\partial)/(\partial x)(e^{-x}sin(x^2+xy))

Lösung

\frac{\partial}{\partial x} (e^{- x} sin (x^{2} + x y))

- e^{- x} sin (x^{2} + x y) + e^{- x} cos (x^{2} + x y) (2 x + y)

Schritte zur Lösung

\frac{\partial}{\partial x} (e^{- x} sin (x^{2} + x y))

Behandle

y

als Konstante

Wende die Produktregel an:

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

f = e^{- x}, g = sin (x^{2} + x y)

= \frac{\partial}{\partial x} (e^{- x}) sin (x^{2} + x y) + \frac{\partial}{\partial x} (sin (x^{2} + x y)) e^{- x}

\frac{\partial}{\partial x} (e^{- x}) = - e^{- x}

\frac{\partial}{\partial x} (sin (x^{2} + x y)) = cos (x^{2} + x y) (2 x + y)

= (- e^{- x}) sin (x^{2} + x y) + cos (x^{2} + x y) (2 x + y) e^{- x}

Vereinfache

= - e^{- x} sin (x^{2} + x y) + e^{- x} cos (x^{2} + x y) (2 x + y)

\frac{\partial}{\partial u} (u^{2} - v^{2})

d er i v a t i v e \frac{1}{1 + t}

d er i v a t i v e \frac{2 x}{x ^{4} + 1}

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

x \to 0 + lim ((1 + 4 x)^{\frac{5}{x}})