Calculus Examples

$e^{r (θ - t)} S^{- 1}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Combine $e^{r (θ - t)}$ and $\frac{1}{S}$ .

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

Step 2.2

Since $\frac{1}{S}$ is constant with respect to $t$ , the derivative of $\frac{e^{r (θ - t)}}{S}$ with respect to $t$ is $\frac{1}{S} \frac{d}{d t} [e^{r (θ - t)}]$ .

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

Step 3

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 $g (t) = r (θ - t)$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $r (θ - t)$ .

$\frac{1}{S} (\frac{d}{d u} [e^{u}] \frac{d}{d t} [r (θ - t)])$

Step 3.2

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

$\frac{1}{S} (e^{u} \frac{d}{d t} [r (θ - t)])$

Step 3.3

Replace all occurrences of $u$ with $r (θ - t)$ .

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

Step 4

Differentiate.

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Step 4.1

Combine $e^{r (θ - t)}$ and $\frac{1}{S}$ .

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

Step 4.2

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

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

Step 4.3

Combine $r$ and $\frac{e^{r (θ - t)}}{S}$ .

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

Step 4.4

By the Sum Rule, the derivative of $θ - t$ with respect to $t$ is $\frac{d}{d t} [θ] + \frac{d}{d t} [- t]$ .

$\frac{r e^{r (θ - t)}}{S} (\frac{d}{d t} [θ] + \frac{d}{d t} [- t])$

Step 4.5

Since $θ$ is constant with respect to $t$ , the derivative of $θ$ with respect to $t$ is $0$ .

$\frac{r e^{r (θ - t)}}{S} (0 + \frac{d}{d t} [- t])$

Step 4.6

Add $0$ and $\frac{d}{d t} [- t]$ .

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

Step 4.7

Since $- 1$ is constant with respect to $t$ , the derivative of $- t$ with respect to $t$ is $- \frac{d}{d t} [t]$ .

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

Step 4.8

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

$\frac{r e^{r (θ - t)}}{S} (- 1 \cdot 1)$

Step 4.9

Combine fractions.

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Step 4.9.1

Multiply $- 1$ by $1$ .

$\frac{r e^{r (θ - t)}}{S} \cdot - 1$

Step 4.9.2

Combine $\frac{r e^{r (θ - t)}}{S}$ and $- 1$ .

$\frac{r e^{r (θ - t)} \cdot - 1}{S}$

Step 4.9.3

Simplify the expression.

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Step 4.9.3.1

Move $- 1$ to the left of $r e^{r (θ - t)}$ .

$\frac{- 1 \cdot (r e^{r (θ - t)})}{S}$

Step 4.9.3.2

Rewrite $- 1 r$ as $- r$ .

$\frac{- r e^{r (θ - t)}}{S}$

Step 4.9.3.3

Move the negative in front of the fraction.

$- \frac{r e^{r (θ - t)}}{S}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- \frac{r e^{r θ + r (- t)}}{S}$

Step 5.2

Rewrite using the commutative property of multiplication.

$- \frac{r e^{r θ - r t}}{S}$