derivative of (ln(x)((x^7+3)/(x^3-10)))

Lösung

\frac{d}{d x} ((ln (x)) (\frac{x ^{7} + 3}{x ^{3} - 10}))

\frac{x ^{7} + 3}{x ( x ^{3} - 10 )} + \frac{ln ( x ) ( 4 x ^{9} - 70 x ^{6} - 9 x ^{2} )}{( x ^{3} - 10 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} ((ln (x)) (\frac{x ^{7} + 3}{x ^{3} - 10}))

Wende die Produktregel an:

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

f = ln (x), g = \frac{x ^{7} + 3}{x ^{3} - 10}

= \frac{d}{d x} (ln (x)) \frac{x ^{7} + 3}{x ^{3} - 10} + \frac{d}{d x} (\frac{x ^{7} + 3}{x ^{3} - 10}) ln (x)

\frac{d}{d x} (ln (x)) = \frac{1}{x}

\frac{d}{d x} (\frac{x ^{7} + 3}{x ^{3} - 10}) = \frac{4 x ^{9} - 70 x ^{6} - 9 x ^{2}}{( x ^{3} - 10 ) ^{2}}

= \frac{1}{x} \cdot \frac{x ^{7} + 3}{x ^{3} - 10} + \frac{4 x ^{9} - 70 x ^{6} - 9 x ^{2}}{( x ^{3} - 10 ) ^{2}} ln (x)

Vereinfache

\frac{1}{x} \cdot \frac{x ^{7} + 3}{x ^{3} - 10} + \frac{4 x ^{9} - 70 x ^{6} - 9 x ^{2}}{( x ^{3} - 10 ) ^{2}} ln (x) : \frac{x ^{7} + 3}{x ( x ^{3} - 10 )} + \frac{ln ( x ) ( 4 x ^{9} - 70 x ^{6} - 9 x ^{2} )}{( x ^{3} - 10 ) ^{2}}

= \frac{x ^{7} + 3}{x ( x ^{3} - 10 )} + \frac{ln ( x ) ( 4 x ^{9} - 70 x ^{6} - 9 x ^{2} )}{( x ^{3} - 10 ) ^{2}}

\int (3 x - 7)^{9} d x

\int \frac{1}{4 x ^{2} - 9} d x

\frac{d}{d x} (3 sin (x) cos (x))

d er i v a t i v e f (x) = x^{- 1} - \frac{1}{x}

\frac{d}{d x} (\frac{2}{3 x - 1})