derivative of sqrt(1-x^2)arccos(x)

Lösung

\frac{d}{d x} (1 - x^{2} arccos (x))

- \frac{x arccos ( x )}{1 - x ^{2}} - 1

Schritte zur Lösung

\frac{d}{d x} (1 - x^{2} arccos (x))

Wende die Produktregel an:

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

f = 1 - x^{2}, g = arccos (x)

= \frac{d}{d x} (1 - x^{2}) arccos (x) + \frac{d}{d x} (arccos (x)) 1 - x^{2}

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

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

= (- \frac{x}{1 - x ^{2}}) arccos (x) + (- \frac{1}{1 - x ^{2}}) 1 - x^{2}

Vereinfache

(- \frac{x}{1 - x ^{2}}) arccos (x) + (- \frac{1}{1 - x ^{2}}) 1 - x^{2} : - \frac{x arccos ( x )}{1 - x ^{2}} - 1

= - \frac{x arccos ( x )}{1 - x ^{2}} - 1

\int cos (e^{t}) d t

\int_{0}^{5} (5 x - x^{2}) d x

\int 9 d x

t an g e n t \frac{4 x}{x + 1}, at (3.3)

x \to - 4 lim (\frac{5}{x + 4})