derivative of e^{xy}(y+x(dy/(dx)))

Lösung

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

e^{x y} (y + x \frac{d y}{d x})^{2} + e^{x y} (2 \frac{d y}{d x} + x \frac{d ^{2} y}{d x ^{2}})

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = e^{x y}, g = y + x \frac{d y}{d x}

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

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

\frac{d}{d x} (y + x \frac{d y}{d x}) = 2 \frac{d y}{d x} + x \frac{d ^{2} y}{d x ^{2}}

= e^{x y} (y + x \frac{d y}{d x}) (y + x \frac{d y}{d x}) + (2 \frac{d y}{d x} + x \frac{d ^{2} y}{d x ^{2}}) e^{x y}

e^{x y} (y + x \frac{d y}{d x}) (y + x \frac{d y}{d x}) = e^{x y} (y + x \frac{d y}{d x})^{2}

= e^{x y} (y + x \frac{d y}{d x})^{2} + e^{x y} (2 \frac{d y}{d x} + x \frac{d ^{2} y}{d x ^{2}})

\int \frac{6}{x ln ( x )} d x

\frac{d}{d x} (x^{5} - 4 x^{9})

d er i v a t i v e G (t) = \frac{1 - 6 t}{2 + t}

n = 1 \sum \infty (- \frac{3}{4})^{n}

x \to 0 - lim (cos (x^{2}))