(\partial)/(\partial x)(x^2e^{sqrt(5xy)})

Lösung

\frac{\partial}{\partial x} (x^{2} e^{5 x y})

\frac{5 e ^{5 x y} x ^{\frac{3}{2}} y}{2} + 2 e^{5 x y} x

Schritte zur Lösung

\frac{\partial}{\partial x} (x^{2} e^{5 x y})

Vereinfache

x^{2} e^{5 x y} : e^{5 x y} x^{2}

= \frac{\partial}{\partial x} (e^{5 x y} x^{2})

Behandle

y

als Konstante

Wende die Produktregel an:

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

f = e^{5 x y}, g = x^{2}

= \frac{\partial}{\partial x} (e^{5 x y}) x^{2} + \frac{\partial}{\partial x} (x^{2}) e^{5 x y}

\frac{\partial}{\partial x} (e^{5 x y}) = \frac{5 y e ^{5 y x}}{2 x}

\frac{\partial}{\partial x} (x^{2}) = 2 x

= \frac{5 y e ^{5 y x}}{2 x} x^{2} + 2 x e^{5 x y}

\frac{5 y e ^{5 y x}}{2 x} x^{2} = \frac{5 e ^{5 x y} x ^{\frac{3}{2}} y}{2}

= \frac{5 e ^{5 x y} x ^{\frac{3}{2}} y}{2} + 2 e^{5 x y} x

x \to 0 - lim (csc (x) - cot (x))

\int \frac{6}{z ^{4}} d z

x \to 4 lim (\frac{x}{x})

t an g e n t f (x) = 4 x^{2} - 2 x, at x = - 1

t an g e n t 3 x^{2}