Calculus Examples

$p e^{r t}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $t$ by $1$ .

$p e^{r t} t$

Step 4

Simplify.

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Step 4.1

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

$e^{r t} p t$

Step 4.2

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

$p t e^{r t}$