Calculus Examples

p e^{r t}

$p e^{r t}$

Step 1

Since

p

$p$ is constant with respect to

r

$r$ , the derivative of

p e^{r t}

$p e^{r t}$ with respect to

r

$r$ is

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

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

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))]

$\frac{d}{d r} [f (g (r))]$ is

f' (g (r)) g' (r)

$f' (g (r)) g' (r)$ where

f (r) = e^{r}

$f (r) = e^{r}$ and

g (r) = r t

$g (r) = r t$ .

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Step 2.1

To apply the Chain Rule, set

u

$u$ as

r t

$r t$ .

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

$\frac{d}{d u} [a^{u}]$ is

a^{u} \ln (a)

$a^{u} \ln (a)$ where

a

$a$ =

e

$e$ .

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

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

Step 2.3

Replace all occurrences of

u

$u$ with

r t

$r t$ .

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

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

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

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

Step 3

Differentiate.

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Step 3.1

Since

t

$t$ is constant with respect to

r

$r$ , the derivative of

r t

$r t$ with respect to

r

$r$ is

t \frac{d}{d r} [r]

$t \frac{d}{d r} [r]$ .

p e^{r t} (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}]

$\frac{d}{d r} [r^{n}]$ is

n r^{n - 1}

$n r^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 3.3

Multiply

t

$t$ by

1

$1$ .

p e^{r t} t

$p e^{r t} t$

p e^{r t} t

$p e^{r t} t$

Step 4

Simplify.

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Step 4.1

Reorder the factors of

p e^{r t} t

$p e^{r t} t$ .

e^{r t} p t

$e^{r t} p t$

Step 4.2

Reorder factors in

e^{r t} p t

$e^{r t} p t$ .

p t e^{r t}

$p t e^{r t}$

p t e^{r t}

$p t e^{r t}$