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Beliebt Algebra >

d/(dt)((x)(a(at^2+bt+c)+b)^{r(x,t)})

Lösung

\frac{d}{d t} ((x) (a (a t^{2} + b t + c) + b)^{r (x, t)})

Lösung

x e^{r (x, t) l n (a (a t^{2} + b t + c) + b)} (\frac{d}{d t} (r (x, t)) ln (a (a t^{2} + b t + c) + b) + \frac{a ( 2 a t + b )}{a ( a t ^{2} + b t + c ) + b} r (x, t))

Schritte zur Lösung

\frac{d}{d t} ((x) (a (a t^{2} + b t + c) + b)^{r (x, t)})

Behandle

a, b, c, x

als Konstante

Entferne die Konstante:

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

= x \frac{d}{d t} ((a (a t^{2} + b t + c) + b)^{r (x, t)})

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

(a (a t^{2} + b t + c) + b)^{r (x, t)} = e^{r (x, t) l n (a (a t^{2} + b t + c) + b)}

= x \frac{d}{d t} (e^{r (x, t) l n (a (a t^{2} + b t + c) + b)})

Wende die Kettenregel an:

e^{r (x, t) l n (a (a t^{2} + b t + c) + b)} \frac{d}{d t} (r (x, t) ln (a (a t^{2} + b t + c) + b))

= e^{r (x, t) l n (a (a t^{2} + b t + c) + b)} \frac{d}{d t} (r (x, t) ln (a (a t^{2} + b t + c) + b))

\frac{d}{d t} (r (x, t) ln (a (a t^{2} + b t + c) + b)) = \frac{d}{d t} (r (x, t)) ln (a (a t^{2} + b t + c) + b) + \frac{a ( 2 a t + b )}{a ( a t ^{2} + b t + c ) + b} r (x, t)

= x e^{r (x, t) l n (a (a t^{2} + b t + c) + b)} (\frac{d}{d t} (r (x, t)) ln (a (a t^{2} + b t + c) + b) + \frac{a ( 2 a t + b )}{a ( a t ^{2} + b t + c ) + b} r (x, t))

Beliebte Beispiele

erweitern (2x-4)/((x^2+x)*(x^2+1))

e x p an d \frac{2 x - 4}{( x ^{2} + x ) \cdot ( x ^{2} + 1 )}

erweitern (5(-2y+3))/6

e x p an d \frac{5 ( - 2 y + 3 )}{6}

vereinfachen (t-sin(t))(sin(t))

s im pl i f y (t - sin (t)) (sin (t))

vereinfachen 6(2^{1/6}-1)

s im pl i f y 6 (2^{\frac{1}{6}} - 1)

erweitern (x+8)/(x^2(x^2+8))

e x p an d \frac{x + 8}{x ^{2} ( x ^{2} + 8 )}