derivative of log_{3}(sin(x^2))

Lösung

\frac{d}{d x} (lo g_{3} (sin (x^{2})))

\frac{2 x cot ( x ^{2} )}{ln ( 3 )}

Schritte zur Lösung

\frac{d}{d x} (lo g_{3} (sin (x^{2})))

Wende die log Regel an:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} (\frac{ln ( sin ( x ^{2} ) )}{ln ( 3 )})

Entferne die Konstante:

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

= \frac{1}{ln ( 3 )} \frac{d}{d x} (ln (sin (x^{2})))

Wende die Kettenregel an:

\frac{1}{sin ( x ^{2} )} \frac{d}{d x} (sin (x^{2}))

= \frac{1}{sin ( x ^{2} )} \frac{d}{d x} (sin (x^{2}))

\frac{d}{d x} (sin (x^{2})) = cos (x^{2}) \cdot 2 x

= \frac{1}{ln ( 3 )} \cdot \frac{1}{sin ( x ^{2} )} cos (x^{2}) \cdot 2 x

Vereinfache

\frac{1}{ln ( 3 )} \cdot \frac{1}{sin ( x ^{2} )} cos (x^{2}) \cdot 2 x : \frac{2 x cot ( x ^{2} )}{ln ( 3 )}

= \frac{2 x cot ( x ^{2} )}{ln ( 3 )}

x \to - 2 + lim (2 - x)

\int x^{2} x + 5 d x

\int in y d y

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

\int 3 e^{\frac{3 t}{2}} t d t