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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

t^{r} = e^{r l n (t)}

= \frac{d}{d t} (e^{r l n (t)})

Apply the chain rule:

e^{r l n (t)} \frac{d}{d t} (r ln (t))

= e^{r l n (t)} \frac{d}{d t} (r ln (t))

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

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

e^{r l n (t)} = t^{r}

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

\int \frac{1}{x ^{2} + 10} d x

\int \frac{2}{- x ^{2} + 5} d x

\frac{s}{s ^{2} + 2 s + 4}

\int + ln (x) d x

\frac{d y}{d x} = \frac{1}{x} (y^{2} + 0.6 y), y (1) = 0.9

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