derivative of x^{1/4}y^{3/4}

Lösung

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

\frac{y + 3 x \frac{d y}{d x}}{4 x ^{\frac{3}{4}} y ^{\frac{1}{4}}}

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x^{\frac{1}{4}}, g = y^{\frac{3}{4}}

= \frac{d}{d x} (x^{\frac{1}{4}}) y^{\frac{3}{4}} + \frac{d}{d x} (y^{\frac{3}{4}}) x^{\frac{1}{4}}

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

\frac{d}{d x} (y^{\frac{3}{4}}) = \frac{3 \frac{d y}{d x}}{4 y ^{\frac{1}{4}}}

= \frac{1}{4 x ^{\frac{3}{4}}} y^{\frac{3}{4}} + \frac{3 \frac{d y}{d x}}{4 y ^{\frac{1}{4}}} x^{\frac{1}{4}}

Vereinfache

\frac{1}{4 x ^{\frac{3}{4}}} y^{\frac{3}{4}} + \frac{3 \frac{d y}{d x}}{4 y ^{\frac{1}{4}}} x^{\frac{1}{4}} : \frac{y + 3 x \frac{d y}{d x}}{4 x ^{\frac{3}{4}} y ^{\frac{1}{4}}}

= \frac{y + 3 x \frac{d y}{d x}}{4 x ^{\frac{3}{4}} y ^{\frac{1}{4}}}

d er i v a t i v e (x^{4} + 7 x^{2} - 9)^{5}

\int (\frac{x + 1}{x - 1}) d x

t an g e n t f (x) = e^{x}, at x = ln (13)

t an g e n t f (x) = e^{x} (8 + 3 x + 5 x^{2}), at x = 0

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