derivative of x^{0.7}y^{0.3}

Lösung

\frac{d}{d x} (x^{0.7} y^{0.3})

\frac{0.7 y ^{0.3}}{x ^{0.3}} + \frac{0.3 x ^{0.7} \frac{d y}{d x}}{y ^{0.7}}

Schritte zur Lösung

\frac{d}{d x} (x^{0.7} y^{0.3})

Wende die Produktregel an:

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

f = x^{0.7}, g = y^{0.3}

= \frac{d}{d x} (x^{0.7}) y^{0.3} + \frac{d}{d x} (y^{0.3}) x^{0.7}

\frac{d}{d x} (x^{0.7}) = \frac{0.7}{x ^{0.3}}

\frac{d}{d x} (y^{0.3}) = \frac{0.3 \frac{d y}{d x}}{y ^{0.7}}

= \frac{0.7}{x ^{0.3}} y^{0.3} + \frac{0.3 \frac{d y}{d x}}{y ^{0.7}} x^{0.7}

Vereinfache

\frac{0.7}{x ^{0.3}} y^{0.3} + \frac{0.3 \frac{d y}{d x}}{y ^{0.7}} x^{0.7} : \frac{0.7 y ^{0.3}}{x ^{0.3}} + \frac{0.3 x ^{0.7} \frac{d y}{d x}}{y ^{0.7}}

= \frac{0.7 y ^{0.3}}{x ^{0.3}} + \frac{0.3 x ^{0.7} \frac{d y}{d x}}{y ^{0.7}}

\frac{d}{d y} (3 cos (3 x) (x - y) + sin (3 x))

n = 1 \sum \infty \frac{n ^{3}}{n ^{3}}

x \to 0 lim (x csc (3 x))

\frac{d}{d x} (ln (4) x^{2} + 3 x + 3)

\int (6 sin (x) + 3 sinh (x)) d x