Calculus Examples

$\frac{- t e^{- t} + e^{- t}}{e^{t}}$

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${(e^{t})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

Move $2$ to the left of $t$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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 $f (t) = t$ and $g (t) = e^{- t}$ .

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

Step 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 $g (t) = - t$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $- 1$ by $1$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 6

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 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Differentiate.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify the expression.

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Step 7.3.1

Multiply $- 1$ by $1$ .

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

Step 7.3.2

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

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

Step 7.3.3

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 9.4

Apply the distributive property.

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

Step 9.5

Simplify the numerator.

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Step 9.5.1

Simplify each term.

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Step 9.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 9.5.1.2

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

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Step 9.5.1.2.1

Move $e^{- t}$ .

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

Step 9.5.1.2.2

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

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

Step 9.5.1.2.3

Add $- t$ and $t$ .

$\frac{- 1 \cdot - 1 e^{0} t + e^{t} (- e^{- t}) + e^{t} (- e^{- t}) - (- t e^{- t}) e^{t} - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.3

Simplify $- 1 \cdot - 1 e^{0} t$ .

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

Step 9.5.1.4

Multiply $- 1$ by $- 1$ .

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

Step 9.5.1.5

Multiply $t$ by $1$ .

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

Step 9.5.1.6

Rewrite using the commutative property of multiplication.

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

Step 9.5.1.7

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

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Step 9.5.1.7.1

Move $e^{- t}$ .

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

Step 9.5.1.7.2

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

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

Step 9.5.1.7.3

Add $- t$ and $t$ .

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

Step 9.5.1.8

Simplify $- e^{0}$ .

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

Step 9.5.1.9

Rewrite using the commutative property of multiplication.

$\frac{t - 1 - e^{t} e^{- t} - (- t e^{- t}) e^{t} - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.10

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

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Step 9.5.1.10.1

Move $e^{- t}$ .

$\frac{t - 1 - (e^{- t} e^{t}) - (- t e^{- t}) e^{t} - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.10.2

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

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

Step 9.5.1.10.3

Add $- t$ and $t$ .

$\frac{t - 1 - e^{0} - (- t e^{- t}) e^{t} - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.11

Simplify $- e^{0}$ .

$\frac{t - 1 - 1 - (- t e^{- t}) e^{t} - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.12

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

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Step 9.5.1.12.1

Move $e^{t}$ .

$\frac{t - 1 - 1 - (- t (e^{t} e^{- t})) - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.12.2

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

$\frac{t - 1 - 1 - (- t e^{t - t}) - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.12.3

Subtract $t$ from $t$ .

$\frac{t - 1 - 1 - (- t e^{0}) - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.13

Simplify $- (- t e^{0})$ .

$\frac{t - 1 - 1 - - t - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.14

Multiply $- - t$ .

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Step 9.5.1.14.1

Multiply $- 1$ by $- 1$ .

$\frac{t - 1 - 1 + 1 t - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.14.2

Multiply $t$ by $1$ .

$\frac{t - 1 - 1 + t - e^{- t} e^{t}}{e^{2 t}}$

Step 9.5.1.15

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

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Step 9.5.1.15.1

Move $e^{t}$ .

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

Step 9.5.1.15.2

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

$\frac{t - 1 - 1 + t - e^{t - t}}{e^{2 t}}$

Step 9.5.1.15.3

Subtract $t$ from $t$ .

$\frac{t - 1 - 1 + t - e^{0}}{e^{2 t}}$

Step 9.5.1.16

Simplify $- e^{0}$ .

$\frac{t - 1 - 1 + t - 1}{e^{2 t}}$

Step 9.5.2

Add $t$ and $t$ .

$\frac{2 t - 1 - 1 - 1}{e^{2 t}}$

Step 9.5.3

Subtract $1$ from $- 1$ .

$\frac{2 t - 2 - 1}{e^{2 t}}$

Step 9.5.4

Subtract $1$ from $- 2$ .

$\frac{2 t - 3}{e^{2 t}}$