derivative 3x(18x^4+(13)/(x+1))

Lösung

ableitung von

3 x (18 x^{4} + \frac{13}{x + 1})

\frac{3 ( 90 x ^{6} + 180 x ^{5} + 90 x ^{4} + 13 )}{( x + 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (3 x (18 x^{4} + \frac{13}{x + 1}))

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d x} (x (18 x^{4} + \frac{13}{x + 1}))

Wende die Produktregel an:

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

f = x, g = 18 x^{4} + \frac{13}{x + 1}

= 3 (\frac{d x}{d x} (18 x^{4} + \frac{13}{x + 1}) + \frac{d}{d x} (18 x^{4} + \frac{13}{x + 1}) x)

\frac{d x}{d x} = 1

\frac{d}{d x} (18 x^{4} + \frac{13}{x + 1}) = 72 x^{3} - \frac{13}{( x + 1 ) ^{2}}

= 3 (1 \cdot (18 x^{4} + \frac{13}{x + 1}) + (72 x^{3} - \frac{13}{( x + 1 ) ^{2}}) x)

Vereinfache

3 (1 \cdot (18 x^{4} + \frac{13}{x + 1}) + (72 x^{3} - \frac{13}{( x + 1 ) ^{2}}) x) : \frac{3 ( 90 x ^{6} + 180 x ^{5} + 90 x ^{4} + 13 )}{( x + 1 ) ^{2}}

= \frac{3 ( 90 x ^{6} + 180 x ^{5} + 90 x ^{4} + 13 )}{( x + 1 ) ^{2}}

(12 y^{2} - x y) d x + x^{2} d y = 0

\int \frac{4}{5 - 3 x} d x

d er i v a t i v e x^{\frac{1}{x}}

\frac{d y}{d t} = 2 (y^{2} - y) t

t an g e n t y = x^{4} - 3 x^{2} + 10, at x = 1