Calculus Examples

$y = \frac{6}{5} - \frac{6}{5} e^{- 20 t}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $\frac{6}{5} - \frac{6}{5} e^{- 20 t}$ with respect to $t$ is $\frac{d}{d t} [\frac{6}{5}] + \frac{d}{d t} [- \frac{6}{5} e^{- 20 t}]$ .

$\frac{d}{d t} [\frac{6}{5}] + \frac{d}{d t} [- \frac{6}{5} e^{- 20 t}]$

Step 1.2

Since $\frac{6}{5}$ is constant with respect to $t$ , the derivative of $\frac{6}{5}$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [- \frac{6}{5} e^{- 20 t}]$

Step 2

Evaluate $\frac{d}{d t} [- \frac{6}{5} e^{- 20 t}]$ .

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Step 2.1

Since $- \frac{6}{5}$ is constant with respect to $t$ , the derivative of $- \frac{6}{5} e^{- 20 t}$ with respect to $t$ is $- \frac{6}{5} \frac{d}{d t} [e^{- 20 t}]$ .

$0 - \frac{6}{5} \frac{d}{d t} [e^{- 20 t}]$

Step 2.2

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 $g (t) = - 20 t$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $- 20 t$ .

$0 - \frac{6}{5} (\frac{d}{d u} [e^{u}] \frac{d}{d t} [- 20 t])$

Step 2.2.2

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

$0 - \frac{6}{5} (e^{u} \frac{d}{d t} [- 20 t])$

Step 2.2.3

Replace all occurrences of $u$ with $- 20 t$ .

$0 - \frac{6}{5} (e^{- 20 t} \frac{d}{d t} [- 20 t])$

Step 2.3

Since $- 20$ is constant with respect to $t$ , the derivative of $- 20 t$ with respect to $t$ is $- 20 \frac{d}{d t} [t]$ .

$0 - \frac{6}{5} (e^{- 20 t} (- 20 \frac{d}{d t} [t]))$

Step 2.4

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

$0 - \frac{6}{5} (e^{- 20 t} (- 20 \cdot 1))$

Step 2.5

Multiply $- 20$ by $1$ .

$0 - \frac{6}{5} (e^{- 20 t} \cdot - 20)$

Step 2.6

Move $- 20$ to the left of $e^{- 20 t}$ .

$0 - \frac{6}{5} (- 20 \cdot e^{- 20 t})$

Step 2.7

Multiply $- 20$ by $- 1$ .

$0 + 20 (\frac{6}{5}) e^{- 20 t}$

Step 2.8

Combine $20$ and $\frac{6}{5}$ .

$0 + \frac{20 \cdot 6}{5} e^{- 20 t}$

Step 2.9

Multiply $20$ by $6$ .

$0 + \frac{120}{5} e^{- 20 t}$

Step 2.10

Combine $\frac{120}{5}$ and $e^{- 20 t}$ .

$0 + \frac{120 e^{- 20 t}}{5}$

Step 2.11

Cancel the common factor of $120$ and $5$ .

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Step 2.11.1

Factor $5$ out of $120 e^{- 20 t}$ .

$0 + \frac{5 (24 e^{- 20 t})}{5}$

Step 2.11.2

Cancel the common factors.

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Step 2.11.2.1

Factor $5$ out of $5$ .

$0 + \frac{5 (24 e^{- 20 t})}{5 (1)}$

Step 2.11.2.2

Cancel the common factor.

$0 + \frac{5 (24 e^{- 20 t})}{5 \cdot 1}$

Step 2.11.2.3

Rewrite the expression.

$0 + \frac{24 e^{- 20 t}}{1}$

Step 2.11.2.4

Divide $24 e^{- 20 t}$ by $1$ .

$0 + 24 e^{- 20 t}$

Step 3

Add $0$ and $24 e^{- 20 t}$ .

$24 e^{- 20 t}$