Calculus Examples

$P = \frac{40}{1 + 11 e^{- 0.08 t}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (P) = \frac{d}{d t} (\frac{40}{1 + 11 e^{- 0.08 t}})$

Step 2

The derivative of $P$ with respect to $t$ is $P'$ .

$P'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 3.1.1

Since $40$ is constant with respect to $t$ , the derivative of $\frac{40}{1 + 11 e^{- 0.08 t}}$ with respect to $t$ is $40 \frac{d}{d t} [\frac{1}{1 + 11 e^{- 0.08 t}}]$ .

$40 \frac{d}{d t} [\frac{1}{1 + 11 e^{- 0.08 t}}]$

Step 3.1.2

Rewrite $\frac{1}{1 + 11 e^{- 0.08 t}}$ as ${(1 + 11 e^{- 0.08 t})}^{- 1}$ .

$40 \frac{d}{d t} [{(1 + 11 e^{- 0.08 t})}^{- 1}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = 1 + 11 e^{- 0.08 t}$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $1 + 11 e^{- 0.08 t}$ .

$40 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d t} [1 + 11 e^{- 0.08 t}])$

Step 3.2.2

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

$40 (- {u_{1}}^{- 2} \frac{d}{d t} [1 + 11 e^{- 0.08 t}])$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $1 + 11 e^{- 0.08 t}$ .

$40 (- {(1 + 11 e^{- 0.08 t})}^{- 2} \frac{d}{d t} [1 + 11 e^{- 0.08 t}])$

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

Multiply $- 1$ by $40$ .

$- 40 ({(1 + 11 e^{- 0.08 t})}^{- 2} \frac{d}{d t} [1 + 11 e^{- 0.08 t}])$

Step 3.3.2

By the Sum Rule, the derivative of $1 + 11 e^{- 0.08 t}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [11 e^{- 0.08 t}]$ .

$- 40 {(1 + 11 e^{- 0.08 t})}^{- 2} (\frac{d}{d t} [1] + \frac{d}{d t} [11 e^{- 0.08 t}])$

Step 3.3.3

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$- 40 {(1 + 11 e^{- 0.08 t})}^{- 2} (0 + \frac{d}{d t} [11 e^{- 0.08 t}])$

Step 3.3.4

Add $0$ and $\frac{d}{d t} [11 e^{- 0.08 t}]$ .

$- 40 {(1 + 11 e^{- 0.08 t})}^{- 2} \frac{d}{d t} [11 e^{- 0.08 t}]$

Step 3.3.5

Since $11$ is constant with respect to $t$ , the derivative of $11 e^{- 0.08 t}$ with respect to $t$ is $11 \frac{d}{d t} [e^{- 0.08 t}]$ .

$- 40 {(1 + 11 e^{- 0.08 t})}^{- 2} (11 \frac{d}{d t} [e^{- 0.08 t}])$

Step 3.3.6

Multiply $11$ by $- 40$ .

$- 440 {(1 + 11 e^{- 0.08 t})}^{- 2} \frac{d}{d t} [e^{- 0.08 t}]$

Step 3.4

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 $g (t) = - 0.08 t$ .

Tap for more steps...

Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $- 0.08 t$ .

$- 440 {(1 + 11 e^{- 0.08 t})}^{- 2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d t} [- 0.08 t])$

Step 3.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$- 440 {(1 + 11 e^{- 0.08 t})}^{- 2} (e^{u_{2}} \frac{d}{d t} [- 0.08 t])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $- 0.08 t$ .

$- 440 {(1 + 11 e^{- 0.08 t})}^{- 2} (e^{- 0.08 t} \frac{d}{d t} [- 0.08 t])$

Step 3.5

Differentiate.

Tap for more steps...

Step 3.5.1

Since $- 0.08$ is constant with respect to $t$ , the derivative of $- 0.08 t$ with respect to $t$ is $- 0.08 \frac{d}{d t} [t]$ .

$- 440 {(1 + 11 e^{- 0.08 t})}^{- 2} (e^{- 0.08 t} (- 0.08 \frac{d}{d t} [t]))$

Step 3.5.2

Multiply $- 0.08$ by $- 440$ .

$35.2 {(1 + 11 e^{- 0.08 t})}^{- 2} (e^{- 0.08 t} (\frac{d}{d t} [t]))$

Step 3.5.3

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

$35.2 {(1 + 11 e^{- 0.08 t})}^{- 2} (e^{- 0.08 t} \cdot 1)$

Step 3.5.4

Multiply $e^{- 0.08 t}$ by $1$ .

$35.2 {(1 + 11 e^{- 0.08 t})}^{- 2} e^{- 0.08 t}$

Step 3.6

Simplify.

Tap for more steps...

Step 3.6.1

Reorder the factors of $35.2 {(1 + 11 e^{- 0.08 t})}^{- 2} e^{- 0.08 t}$ .

$35.2 e^{- 0.08 t} {(1 + 11 e^{- 0.08 t})}^{- 2}$

Step 3.6.2

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

$35.2 e^{- 0.08 t} \frac{1}{{(1 + 11 e^{- 0.08 t})}^{2}}$

Step 3.6.3

Multiply $35.2 e^{- 0.08 t} \frac{1}{{(1 + 11 e^{- 0.08 t})}^{2}}$ .

Tap for more steps...

Step 3.6.3.1

Combine $\frac{1}{{(1 + 11 e^{- 0.08 t})}^{2}}$ and $35.2$ .

$\frac{35.2}{{(1 + 11 e^{- 0.08 t})}^{2}} e^{- 0.08 t}$

Step 3.6.3.2

Combine $\frac{35.2}{{(1 + 11 e^{- 0.08 t})}^{2}}$ and $e^{- 0.08 t}$ .

$\frac{35.2 e^{- 0.08 t}}{{(1 + 11 e^{- 0.08 t})}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$P' = \frac{35.2 e^{- 0.08 t}}{{(1 + 11 e^{- 0.08 t})}^{2}}$

Step 5

Replace $P'$ with $\frac{d P}{d t}$ .

$\frac{d P}{d t} = \frac{35.2 e^{- 0.08 t}}{{(1 + 11 e^{- 0.08 t})}^{2}}$