Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{3}$ and $g (t) = e^{- t} + e^{t}$ .

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Step 1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d t} [e^{- t} + e^{t}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$3 {u_{1}}^{2} \frac{d}{d t} [e^{- t} + e^{t}]$

Step 1.3

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

$3 {(e^{- t} + e^{t})}^{2} \frac{d}{d t} [e^{- t} + e^{t}]$

Step 2

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

$3 {(e^{- t} + e^{t})}^{2} (\frac{d}{d t} [e^{- t}] + \frac{d}{d t} [e^{t}])$

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 $g (t) = - t$ .

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Step 3.1

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

$3 {(e^{- t} + e^{t})}^{2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d t} [- t] + \frac{d}{d t} [e^{t}])$

Step 3.2

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

$3 {(e^{- t} + e^{t})}^{2} (e^{u_{2}} \frac{d}{d t} [- t] + \frac{d}{d t} [e^{t}])$

Step 3.3

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

$3 {(e^{- t} + e^{t})}^{2} (e^{- t} \frac{d}{d t} [- t] + \frac{d}{d t} [e^{t}])$

Step 4

Differentiate.

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Step 4.1

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

$3 {(e^{- t} + e^{t})}^{2} (e^{- t} (- \frac{d}{d t} [t]) + \frac{d}{d t} [e^{t}])$

Step 4.2

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

$3 {(e^{- t} + e^{t})}^{2} (e^{- t} (- 1 \cdot 1) + \frac{d}{d t} [e^{t}])$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $- 1$ by $1$ .

$3 {(e^{- t} + e^{t})}^{2} (e^{- t} \cdot - 1 + \frac{d}{d t} [e^{t}])$

Step 4.3.2

Move $- 1$ to the left of $e^{- t}$ .

$3 {(e^{- t} + e^{t})}^{2} (- 1 \cdot e^{- t} + \frac{d}{d t} [e^{t}])$

Step 4.3.3

Rewrite $- 1 e^{- t}$ as $- e^{- t}$ .

$3 {(e^{- t} + e^{t})}^{2} (- e^{- t} + \frac{d}{d t} [e^{t}])$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Rewrite ${(e^{- t} + e^{t})}^{2}$ as $(e^{- t} + e^{t}) (e^{- t} + e^{t})$ .

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

Step 6.2

Expand $(e^{- t} + e^{t}) (e^{- t} + e^{t})$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

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

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

Step 6.3.1.1.2

Subtract $t$ from $- t$ .

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

Step 6.3.1.2

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

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Step 6.3.1.2.1

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

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

Step 6.3.1.2.2

Add $- t$ and $t$ .

$3 (e^{- 2 t} + e^{0} + e^{t} e^{- t} + e^{t} e^{t}) (- e^{- t} + e^{t})$

Step 6.3.1.3

Simplify $e^{0}$ .

$3 (e^{- 2 t} + 1 + e^{t} e^{- t} + e^{t} e^{t}) (- e^{- t} + e^{t})$

Step 6.3.1.4

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

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Step 6.3.1.4.1

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

$3 (e^{- 2 t} + 1 + e^{t - t} + e^{t} e^{t}) (- e^{- t} + e^{t})$

Step 6.3.1.4.2

Subtract $t$ from $t$ .

$3 (e^{- 2 t} + 1 + e^{0} + e^{t} e^{t}) (- e^{- t} + e^{t})$

Step 6.3.1.5

Simplify $e^{0}$ .

$3 (e^{- 2 t} + 1 + 1 + e^{t} e^{t}) (- e^{- t} + e^{t})$

Step 6.3.1.6

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

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Step 6.3.1.6.1

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

$3 (e^{- 2 t} + 1 + 1 + e^{t + t}) (- e^{- t} + e^{t})$

Step 6.3.1.6.2

Add $t$ and $t$ .

$3 (e^{- 2 t} + 1 + 1 + e^{2 t}) (- e^{- t} + e^{t})$

Step 6.3.2

Add $1$ and $1$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $3$ by $2$ .

$(3 e^{- 2 t} + 6 + 3 e^{2 t}) (- e^{- t} + e^{t})$

Step 6.6

Expand $(3 e^{- 2 t} + 6 + 3 e^{2 t}) (- e^{- t} + e^{t})$ by multiplying each term in the first expression by each term in the second expression.

$3 e^{- 2 t} (- e^{- t}) + 3 e^{- 2 t} e^{t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7

Simplify each term.

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Step 6.7.1

Rewrite using the commutative property of multiplication.

$3 \cdot - 1 e^{- 2 t} e^{- t} + 3 e^{- 2 t} e^{t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.2

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

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Step 6.7.2.1

Move $e^{- t}$ .

$3 \cdot - 1 (e^{- t} e^{- 2 t}) + 3 e^{- 2 t} e^{t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.2.2

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

$3 \cdot - 1 e^{- t - 2 t} + 3 e^{- 2 t} e^{t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.2.3

Subtract $2 t$ from $- t$ .

$3 \cdot - 1 e^{- 3 t} + 3 e^{- 2 t} e^{t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.3

Multiply $3$ by $- 1$ .

$- 3 e^{- 3 t} + 3 e^{- 2 t} e^{t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.4

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

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Step 6.7.4.1

Move $e^{t}$ .

$- 3 e^{- 3 t} + 3 (e^{t} e^{- 2 t}) + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.4.2

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

$- 3 e^{- 3 t} + 3 e^{t - 2 t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.4.3

Subtract $2 t$ from $t$ .

$- 3 e^{- 3 t} + 3 e^{- t} + 6 (- e^{- t}) + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.5

Multiply $- 1$ by $6$ .

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} + 3 e^{2 t} (- e^{- t}) + 3 e^{2 t} e^{t}$

Step 6.7.6

Rewrite using the commutative property of multiplication.

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} + 3 \cdot - 1 e^{2 t} e^{- t} + 3 e^{2 t} e^{t}$

Step 6.7.7

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

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Step 6.7.7.1

Move $e^{- t}$ .

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} + 3 \cdot - 1 (e^{- t} e^{2 t}) + 3 e^{2 t} e^{t}$

Step 6.7.7.2

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

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} + 3 \cdot - 1 e^{- t + 2 t} + 3 e^{2 t} e^{t}$

Step 6.7.7.3

Add $- t$ and $2 t$ .

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} + 3 \cdot - 1 e^{t} + 3 e^{2 t} e^{t}$

Step 6.7.8

Multiply $3$ by $- 1$ .

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} - 3 e^{t} + 3 e^{2 t} e^{t}$

Step 6.7.9

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

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Step 6.7.9.1

Move $e^{t}$ .

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} - 3 e^{t} + 3 (e^{t} e^{2 t})$

Step 6.7.9.2

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

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} - 3 e^{t} + 3 e^{t + 2 t}$

Step 6.7.9.3

Add $t$ and $2 t$ .

$- 3 e^{- 3 t} + 3 e^{- t} - 6 e^{- t} + 6 e^{t} - 3 e^{t} + 3 e^{3 t}$

Step 6.8

Subtract $6 e^{- t}$ from $3 e^{- t}$ .

$- 3 e^{- 3 t} - 3 e^{- t} + 6 e^{t} - 3 e^{t} + 3 e^{3 t}$

Step 6.9

Subtract $3 e^{t}$ from $6 e^{t}$ .

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