Calculus Examples

$e^{t} (e^{2 t} - e^{- 2 t})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = e^{t}$ and $g (t) = e^{2 t} - e^{- 2 t}$ .

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

Step 2

By the Sum Rule, the derivative of $e^{2 t} - e^{- 2 t}$ with respect to $t$ is $\frac{d}{d t} [e^{2 t}] + \frac{d}{d t} [- e^{- 2 t}]$ .

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

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 $g (t) = 2 t$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $2 t$ .

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

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3

Replace all occurrences of $u_{1}$ with $2 t$ .

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

Step 4

Differentiate.

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Step 4.1

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

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

Step 4.2

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

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $2$ by $1$ .

$e^{t} (e^{2 t} \cdot 2 + \frac{d}{d t} [- e^{- 2 t}]) + (e^{2 t} - e^{- 2 t}) \frac{d}{d t} [e^{t}]$

Step 4.3.2

Move $2$ to the left of $e^{2 t}$ .

$e^{t} (2 \cdot e^{2 t} + \frac{d}{d t} [- e^{- 2 t}]) + (e^{2 t} - e^{- 2 t}) \frac{d}{d t} [e^{t}]$

Step 4.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- e^{- 2 t}$ with respect to $t$ is $- \frac{d}{d t} [e^{- 2 t}]$ .

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

Step 5

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 $g (t) = - 2 t$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Differentiate.

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Step 6.1

Since $- 2$ is constant with respect to $t$ , the derivative of $- 2 t$ with respect to $t$ is $- 2 \frac{d}{d t} [t]$ .

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

Step 6.2

Multiply $- 2$ by $- 1$ .

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

Step 6.3

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

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

Step 6.4

Multiply $2$ by $1$ .

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

Step 7

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

$e^{t} (2 e^{2 t} + 2 e^{- 2 t}) + (e^{2 t} - e^{- 2 t}) e^{t}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$e^{t} (2 e^{2 t}) + e^{t} (2 e^{- 2 t}) + (e^{2 t} - e^{- 2 t}) e^{t}$

Step 8.2

Apply the distributive property.

$e^{t} (2 e^{2 t}) + e^{t} (2 e^{- 2 t}) + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3

Combine terms.

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Step 8.3.1

Multiply $e^{t}$ by $e^{2 t}$ by adding the exponents.

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Step 8.3.1.1

Move $e^{2 t}$ .

$e^{2 t} e^{t} \cdot 2 + e^{t} (2 e^{- 2 t}) + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{2 t + t} \cdot 2 + e^{t} (2 e^{- 2 t}) + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.1.3

Add $2 t$ and $t$ .

$e^{3 t} \cdot 2 + e^{t} (2 e^{- 2 t}) + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.2

Move $2$ to the left of $e^{3 t}$ .

$2 \cdot e^{3 t} + e^{t} (2 e^{- 2 t}) + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.3

Multiply $e^{t}$ by $e^{- 2 t}$ by adding the exponents.

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Step 8.3.3.1

Move $e^{- 2 t}$ .

$2 e^{3 t} + e^{- 2 t} e^{t} \cdot 2 + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 e^{3 t} + e^{- 2 t + t} \cdot 2 + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.3.3

Add $- 2 t$ and $t$ .

$2 e^{3 t} + e^{- t} \cdot 2 + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.4

Move $2$ to the left of $e^{- t}$ .

$2 e^{3 t} + 2 \cdot e^{- t} + e^{2 t} e^{t} - e^{- 2 t} e^{t}$

Step 8.3.5

Multiply $e^{2 t}$ by $e^{t}$ by adding the exponents.

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Step 8.3.5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 e^{3 t} + 2 e^{- t} + e^{2 t + t} - e^{- 2 t} e^{t}$

Step 8.3.5.2

Add $2 t$ and $t$ .

$2 e^{3 t} + 2 e^{- t} + e^{3 t} - e^{- 2 t} e^{t}$

Step 8.3.6

Multiply $e^{- 2 t}$ by $e^{t}$ by adding the exponents.

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Step 8.3.6.1

Move $e^{t}$ .

$2 e^{3 t} + 2 e^{- t} + e^{3 t} - (e^{t} e^{- 2 t})$

Step 8.3.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 e^{3 t} + 2 e^{- t} + e^{3 t} - e^{t - 2 t}$

Step 8.3.6.3

Subtract $2 t$ from $t$ .

$2 e^{3 t} + 2 e^{- t} + e^{3 t} - e^{- t}$

Step 8.3.7

Add $2 e^{3 t}$ and $e^{3 t}$ .

$3 e^{3 t} + 2 e^{- t} - e^{- t}$

Step 8.3.8

Subtract $e^{- t}$ from $2 e^{- t}$ .

$3 e^{3 t} + e^{- t}$