(\partial)/(\partial x)(x/(sin(2x+3y+z^2)))

Lösung

\frac{\partial}{\partial x} (\frac{x}{sin ( 2 x + 3 y + z ^{2} )})

csc (2 x + 3 y + z^{2}) - 2 x cot (2 x + 3 y + z^{2}) csc (2 x + 3 y + z^{2})

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{x}{sin ( 2 x + 3 y + z ^{2} )})

Vereinfache

\frac{x}{sin ( 2 x + 3 y + z ^{2} )} : x csc (2 x + 3 y + z^{2})

= \frac{\partial}{\partial x} (x csc (2 x + 3 y + z^{2}))

Behandle

y, z

als Konstanten

Wende die Produktregel an:

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

f = x, g = csc (2 x + 3 y + z^{2})

= \frac{\partial}{\partial x} (x) csc (2 x + 3 y + z^{2}) + \frac{\partial}{\partial x} (csc (2 x + 3 y + z^{2})) x

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (csc (2 x + 3 y + z^{2})) = - cot (2 x + 3 y + z^{2}) csc (2 x + 3 y + z^{2}) \cdot 2

= 1 \cdot csc (2 x + 3 y + z^{2}) + (- cot (2 x + 3 y + z^{2}) csc (2 x + 3 y + z^{2}) \cdot 2) x

Vereinfache

= csc (2 x + 3 y + z^{2}) - 2 x cot (2 x + 3 y + z^{2}) csc (2 x + 3 y + z^{2})

\frac{\partial}{\partial u} (3 v cos (u))

\frac{d}{d x} (ln ((x^{2} + 4)^{5}))

x \to 4 lim (\frac{( 2 )}{x - 4})

x \to 2 - lim (\frac{2 x}{x ^{2} - 4})

y^{^{''}} + 8 y^{^{'}} + 25 y = 0, y (0) = 3, y^{^{'}} (0) = - 10