Calculus Examples

$P e^{r t}$

Step 1

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

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

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 $g (t) = r t$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $r$ by $1$ .

$P e^{r t} r$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Reorder the factors of $P e^{r t} r$ .

$e^{r t} P r$

Step 4.2

Reorder factors in $e^{r t} P r$ .

$P r e^{r t}$