Calculus Examples

$f (t) = \frac{t}{e^{t}}$

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

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

Step 2

Differentiate using the Power Rule.

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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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Factor $e^{t}$ out of $- e^{t} t + e^{t}$ .

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Step 4.2.1

Factor $e^{t}$ out of $- e^{t} t$ .

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

Step 4.2.2

Multiply by $1$ .

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

Step 4.2.3

Factor $e^{t}$ out of $e^{t} (- t) + e^{t} \cdot 1$ .

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

Step 4.3

Cancel the common factor of $e^{t}$ and $e^{2 t}$ .

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Step 4.3.1

Factor $e^{2 t}$ out of $e^{t} (- t + 1)$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Multiply by $1$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.3.2.4

Divide $e^{- t} (- t + 1)$ by $1$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Rewrite using the commutative property of multiplication.

$- e^{- t} t + e^{- t} \cdot 1$

Step 4.6

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

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

Step 4.7

Reorder factors in $- e^{- t} t + e^{- t}$ .

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