derivative of ycos(xln(y))

Lösung

\frac{d}{d x} (y cos (x ln (y)))

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

Schritte zur Lösung

\frac{d}{d x} (y cos (x ln (y)))

Wende die Produktregel an:

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

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

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

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

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

Vereinfache

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

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

s im pl i f y \frac{x}{2 - x}

\frac{d y}{d x} = 5 (1 - \frac{x}{10})

\int_{- 2}^{0} - x + 2 d x

\int - 20 x^{- 5} d x

\frac{d y}{d x} = e^{y - x}