Calculus Examples

$y = \ln (3 t e^{- t})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $3 t e^{- t}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d t} [3 t e^{- t}]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $3 t e^{- t}$ .

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

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t e^{- t}$ with respect to $t$ is $3 \frac{d}{d t} [t e^{- t}]$ .

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

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $3$ and $\frac{1}{3 t e^{- t}}$ .

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

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

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

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

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 $u_{2}$ as $- t$ .

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

Step 4.2

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

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

Step 4.3

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

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

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 $- \frac{d}{d t} [t]$ .

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

Step 5.2

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

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

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $- 1$ by $1$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Combine terms.

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Step 6.2.1

Combine $t$ and $\frac{1}{t e^{- t}}$ .

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

Step 6.2.2

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

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

Step 6.2.3

Cancel the common factor of $t$ .

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Step 6.2.3.1

Cancel the common factor.

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

Step 6.2.3.2

Rewrite the expression.

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

Step 6.2.4

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

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Step 6.2.4.1

Cancel the common factor.

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

Step 6.2.4.2

Rewrite the expression.

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

Step 6.2.5

Multiply $- 1$ by $1$ .

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

Step 6.2.6

Combine $\frac{1}{t e^{- t}}$ and $e^{- t}$ .

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

Step 6.2.7

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

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Step 6.2.7.1

Cancel the common factor.

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

Step 6.2.7.2

Rewrite the expression.

$- 1 + \frac{1}{t}$

Step 6.3

Reorder terms.

$\frac{1}{t} - 1$