What is the d/(dt)((r)(t^r)) ?

The d/(dt)((r)(t^r)) is (dr)/(dt)t^r+t^r(ln(t)(dr)/(dt)+r/t)r

What is the first d/(dt)((r)(t^r)) ?

The first d/(dt)((r)(t^r)) is (dr)/(dt)t^r+t^r(ln(t)(dr)/(dt)+r/t)r

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\frac{d}{d t} ((r) (t^{r}))

\frac{d r}{d t} t^{r} + t^{r} (ln (t) \frac{d r}{d t} + \frac{r}{t}) r

Solution steps

\frac{d}{d t} ((r) (t^{r}))

Apply the Product Rule:

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

f = r, g = t^{r}

= \frac{d r}{d t} t^{r} + \frac{d}{d t} (t^{r}) r

\frac{d}{d t} (t^{r}) = t^{r} (ln (t) \frac{d r}{d t} + \frac{r}{t})

= \frac{d r}{d t} t^{r} + t^{r} (ln (t) \frac{d r}{d t} + \frac{r}{t}) r

2^{2 x} + 3 (2^{x}) - 18

1 + (sin (θ))^{2}

x^{2} - 0.25

(x - 3)^{3} + 125

e^{- 2 t} u (t) + e^{3 t} u (- t)

What is the d/(dt)((r)(t^r)) ?
The d/(dt)((r)(t^r)) is (dr)/(dt)t^r+t^r(ln(t)(dr)/(dt)+r/t)r
What is the first d/(dt)((r)(t^r)) ?
The first d/(dt)((r)(t^r)) is (dr)/(dt)t^r+t^r(ln(t)(dr)/(dt)+r/t)r