derivative of e^ycos(xy)

Lösung

\frac{d}{d x} (e^{y} cos (x y))

e^{y} cos (x y) \frac{d y}{d x} - e^{y} sin (x y) (y + x \frac{d y}{d x})

Schritte zur Lösung

\frac{d}{d x} (e^{y} cos (x y))

Wende die Produktregel an:

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

f = e^{y}, g = cos (x y)

= \frac{d}{d x} (e^{y}) cos (x y) + \frac{d}{d x} (cos (x y)) e^{y}

\frac{d}{d x} (e^{y}) = e^{y} \frac{d y}{d x}

\frac{d}{d x} (cos (x y)) = - sin (x y) (y + x \frac{d y}{d x})

= e^{y} \frac{d y}{d x} cos (x y) + (- sin (x y) (y + x \frac{d y}{d x})) e^{y}

Vereinfache

= e^{y} cos (x y) \frac{d y}{d x} - e^{y} sin (x y) (y + x \frac{d y}{d x})

t an g e n t y = (x^{3} - 25 x)^{8}, (- 5, 0)

\frac{d}{d x} (\frac{x ^{4} - 3 x ^{2}}{( x ^{2} - 1 ) ^{2}})

a re a x = \frac{( y - 2 ) ^{2}}{2}, y = 6 - x

t an g e n t f (x) = \frac{x}{( 3 x - 2 ) ^{6}}, at x = 1

\int x^{5} e^{x^{3} - 7} d x