(\partial)/(\partial x)(sin(x)ln(x^4+1/x))

Lösung

\frac{\partial}{\partial x} (sin (x) ln (x^{4} + \frac{1}{x}))

cos (x) ln (x^{4} + \frac{1}{x}) + \frac{sin ( x ) ( 4 x ^{5} - 1 )}{x ( x ^{5} + 1 )}

Schritte zur Lösung

\frac{\partial}{\partial x} (sin (x) ln (x^{4} + \frac{1}{x}))

Wende die Produktregel an:

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

f = sin (x), g = ln (x^{4} + \frac{1}{x})

= \frac{\partial}{\partial x} (sin (x)) ln (x^{4} + \frac{1}{x}) + \frac{\partial}{\partial x} (ln (x^{4} + \frac{1}{x})) sin (x)

\frac{\partial}{\partial x} (sin (x)) = cos (x)

\frac{\partial}{\partial x} (ln (x^{4} + \frac{1}{x})) = \frac{4 x ^{5} - 1}{x ( x ^{5} + 1 )}

= cos (x) ln (x^{4} + \frac{1}{x}) + \frac{4 x ^{5} - 1}{x ( x ^{5} + 1 )} sin (x)

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= cos (x) ln (x^{4} + \frac{1}{x}) + \frac{sin ( x ) ( 4 x ^{5} - 1 )}{x ( x ^{5} + 1 )}

e^{y} + y cos (x) + (x e^{y} + sin (x)) \frac{d y}{d x} = 0

\int 4 x^{3} - 3 x^{2} + 6 x - 1 d x

\frac{d}{d x} (e^{5 x y})

\int 14 (x - 5)^{6} d x

\frac{\partial}{\partial θ} (z)