derivative of 1/15 cos^3(x(3cos^2(x)-5))

Lösung

\frac{d}{d x} (\frac{1}{15} cos^{3} (x) (3 cos^{2} (x) - 5))

- \frac{cos ^{2} ( x ) ( sin ( x ) ( 3 cos ^{2} ( x ) - 5 ) + sin ( 2 x ) cos ( x ) )}{5}

Schritte zur Lösung

\frac{d}{d x} (\frac{1}{15} cos^{3} (x) (3 cos^{2} (x) - 5))

Entferne die Konstante:

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

= \frac{1}{15} \frac{d}{d x} (cos^{3} (x) (3 cos^{2} (x) - 5))

Wende die Produktregel an:

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

f = cos^{3} (x), g = 3 cos^{2} (x) - 5

= \frac{1}{15} (\frac{d}{d x} (cos^{3} (x)) (3 cos^{2} (x) - 5) + \frac{d}{d x} (3 cos^{2} (x) - 5) cos^{3} (x))

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

\frac{d}{d x} (3 cos^{2} (x) - 5) = - 3 sin (2 x)

= \frac{1}{15} ((- 3 cos^{2} (x) sin (x)) (3 cos^{2} (x) - 5) + (- 3 sin (2 x)) cos^{3} (x))

Vereinfache

\frac{1}{15} ((- 3 cos^{2} (x) sin (x)) (3 cos^{2} (x) - 5) + (- 3 sin (2 x)) cos^{3} (x)) : - \frac{cos ^{2} ( x ) ( sin ( x ) ( 3 cos ^{2} ( x ) - 5 ) + sin ( 2 x ) cos ( x ) )}{5}

= - \frac{cos ^{2} ( x ) ( sin ( x ) ( 3 cos ^{2} ( x ) - 5 ) + sin ( 2 x ) cos ( x ) )}{5}

\int_{4}^{e^{5}} \frac{1}{x} d x

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

d er i v a t i v e y = \frac{4 x}{e ^{x}}

\int_{- 1}^{x^{3}} t^{5} d t

\int 3 tan^{2} (x) sec^{2} (x) d x