Calculus Examples

$r e^{r t}$

Step 1

Since $r$ is constant with respect to $t$ , the derivative of $r e^{r t}$ with respect to $t$ is $r \frac{d}{d t} [e^{r t}]$ .

$r \frac{d}{d t} [e^{r t}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = r t$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $r t$ .

$r (\frac{d}{d u} [e^{u}] \frac{d}{d t} [r t])$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$r (e^{u} \frac{d}{d t} [r t])$

Step 2.3

Replace all occurrences of $u$ with $r t$ .

$r (e^{r t} \frac{d}{d t} [r t])$

Step 3

Since $r$ is constant with respect to $t$ , the derivative of $r t$ with respect to $t$ is $r \frac{d}{d t} [t]$ .

$r (e^{r t} (r \frac{d}{d t} [t]))$

Step 4

Raise $r$ to the power of $1$ .

$r^{1} r (e^{r t} (\frac{d}{d t} [t]))$

Step 5

Raise $r$ to the power of $1$ .

$r^{1} r^{1} (e^{r t} (\frac{d}{d t} [t]))$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r^{1 + 1} (e^{r t} (\frac{d}{d t} [t]))$

Step 7

Add $1$ and $1$ .

$r^{2} (e^{r t} (\frac{d}{d t} [t]))$

Step 8

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$r^{2} (e^{r t} \cdot 1)$

Step 9

Multiply $e^{r t}$ by $1$ .

$r^{2} e^{r t}$