derivative of (sin(x^2)/(sin^2(x)))

Lösung

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

- 2 csc^{2} (x) cot (x) sin (x^{2}) + 2 x csc^{2} (x) cos (x^{2})

Schritte zur Lösung

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

Vereinfache

\frac{sin ( x ^{2} )}{sin ^{2} ( x )} : csc^{2} (x) sin (x^{2})

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

Wende die Produktregel an:

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

f = csc^{2} (x), g = sin (x^{2})

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

\frac{d}{d x} (csc^{2} (x)) = - 2 csc^{2} (x) cot (x)

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

= (- 2 csc^{2} (x) cot (x)) sin (x^{2}) + cos (x^{2}) \cdot 2 x csc^{2} (x)

Vereinfache

= - 2 csc^{2} (x) cot (x) sin (x^{2}) + 2 x csc^{2} (x) cos (x^{2})

d er i v a t i v e f (x) = 2^{20}

d er i v a t i v e f (x) = 2 - x

\frac{d}{d x} (\frac{( 3 x ^{2} - 5 ) ^{4}}{( 4 x ^{3} + 2 ) ^{5}})

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

\int 16 - 4 x^{2} d x