derivative of 8cosh(x/4 ln(arcsin(x)))

Lösung

\frac{d}{d x} (8 cosh (\frac{x}{4}) ln (arcsin (x)))

8 (\frac{1}{4} sinh (\frac{x}{4}) ln (arcsin (x)) + \frac{cosh ( \frac{x}{4} )}{arcsin ( x ) 1 - x ^{2}})

Schritte zur Lösung

\frac{d}{d x} (8 cosh (\frac{x}{4}) ln (arcsin (x)))

Entferne die Konstante:

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

= 8 \frac{d}{d x} (cosh (\frac{x}{4}) ln (arcsin (x)))

Wende die Produktregel an:

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

f = cosh (\frac{x}{4}), g = ln (arcsin (x))

= 8 (\frac{d}{d x} (cosh (\frac{x}{4})) ln (arcsin (x)) + \frac{d}{d x} (ln (arcsin (x))) cosh (\frac{x}{4}))

\frac{d}{d x} (cosh (\frac{x}{4})) = sinh (\frac{x}{4}) \frac{1}{4}

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

= 8 (sinh (\frac{x}{4}) \frac{1}{4} ln (arcsin (x)) + \frac{1}{arcsin ( x ) 1 - x ^{2}} cosh (\frac{x}{4}))

\frac{1}{arcsin ( x ) 1 - x ^{2}} cosh (\frac{x}{4}) = \frac{cosh ( \frac{x}{4} )}{arcsin ( x ) 1 - x ^{2}}

= 8 (\frac{1}{4} sinh (\frac{x}{4}) ln (arcsin (x)) + \frac{cosh ( \frac{x}{4} )}{arcsin ( x ) 1 - x ^{2}})

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