derivative of csc^3(pixarccot(sin(x)))

Lösung

\frac{d}{d x} (csc^{3} (π x) arccot (sin (x)))

- 3 π csc^{3} (π x) cot (π x) arccot (sin (x)) - \frac{csc ^{3} ( π x ) cos ( x )}{sin ^{2} ( x ) + 1}

Schritte zur Lösung

\frac{d}{d x} (csc^{3} (π x) arccot (sin (x)))

Wende die Produktregel an:

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

f = csc^{3} (π x), g = arccot (sin (x))

= \frac{d}{d x} (csc^{3} (π x)) arccot (sin (x)) + \frac{d}{d x} (arccot (sin (x))) csc^{3} (π x)

\frac{d}{d x} (csc^{3} (π x)) = - 3 π csc^{3} (π x) cot (π x)

\frac{d}{d x} (arccot (sin (x))) = - \frac{cos ( x )}{sin ^{2} ( x ) + 1}

= (- 3 π csc^{3} (π x) cot (π x)) arccot (sin (x)) + (- \frac{cos ( x )}{sin ^{2} ( x ) + 1}) csc^{3} (π x)

Vereinfache

(- 3 π csc^{3} (π x) cot (π x)) arccot (sin (x)) + (- \frac{cos ( x )}{sin ^{2} ( x ) + 1}) csc^{3} (π x) : - 3 π csc^{3} (π x) cot (π x) arccot (sin (x)) - \frac{csc ^{3} ( π x ) cos ( x )}{sin ^{2} ( x ) + 1}

= - 3 π csc^{3} (π x) cot (π x) arccot (sin (x)) - \frac{csc ^{3} ( π x ) cos ( x )}{sin ^{2} ( x ) + 1}

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