derivative of cos(x^2(2x))

Lösung

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

2 (- 2 x^{2} sin (x^{2}) + cos (x^{2}))

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

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

= 2 (\frac{d}{d x} (cos (x^{2})) x + \frac{d x}{d x} cos (x^{2}))

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

\frac{d x}{d x} = 1

= 2 ((- sin (x^{2}) \cdot 2 x) x + 1 \cdot cos (x^{2}))

Vereinfache

2 ((- sin (x^{2}) \cdot 2 x) x + 1 \cdot cos (x^{2})) : 2 (- 2 x^{2} sin (x^{2}) + cos (x^{2}))

= 2 (- 2 x^{2} sin (x^{2}) + cos (x^{2}))

d er i v a t i v e 5 x^{2} - 3 x + 1

\frac{d}{d x} (3 \cdot 5^{x})

x \to 3 lim (2 x^{3} - 10 x - 8)

x \to 0 lim (\frac{x ^{2} - 9}{x - 3})

x \to 0 lim (\frac{( 3 + x ) ^{2} - 9}{x})