derivative f(x)=7x^4arctan(4x^4)

Lösung

ableitung von

f (x) = 7 x^{4} arctan (4 x^{4})

7 (4 x^{3} arctan (4 x^{4}) + \frac{16 x ^{7}}{16 x ^{8} + 1})

Schritte zur Lösung

\frac{d}{d x} (7 x^{4} arctan (4 x^{4}))

Entferne die Konstante:

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

= 7 \frac{d}{d x} (x^{4} arctan (4 x^{4}))

Wende die Produktregel an:

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

f = x^{4}, g = arctan (4 x^{4})

= 7 (\frac{d}{d x} (x^{4}) arctan (4 x^{4}) + \frac{d}{d x} (arctan (4 x^{4})) x^{4})

\frac{d}{d x} (x^{4}) = 4 x^{3}

\frac{d}{d x} (arctan (4 x^{4})) = \frac{16 x ^{3}}{16 x ^{8} + 1}

= 7 (4 x^{3} arctan (4 x^{4}) + \frac{16 x ^{3}}{16 x ^{8} + 1} x^{4})

\frac{16 x ^{3}}{16 x ^{8} + 1} x^{4} = \frac{16 x ^{7}}{16 x ^{8} + 1}

= 7 (4 x^{3} arctan (4 x^{4}) + \frac{16 x ^{7}}{16 x ^{8} + 1})

\int \frac{7}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x

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

x \to 1 lim (4 x^{3} - x^{2} + 7 x)

\frac{d}{d x} (\frac{1 + csc ( x )}{1 - csc ( x )})

\int cos^{2} (4 x) sin^{3} (4 x) d x