Calculus Examples

$t^{12} \ln (| t |)$

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 $f (t) = t^{12}$ and $g (t) = \ln (| t |)$ .

$t^{12} \frac{d}{d t} [\ln (| t |)] + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 2

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 $g (t) = | t |$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $| t |$ .

$t^{12} (\frac{d}{d u} [\ln (u)] \frac{d}{d t} [| t |]) + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 2.2

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

$t^{12} (\frac{1}{u} \frac{d}{d t} [| t |]) + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 2.3

Replace all occurrences of $u$ with $| t |$ .

$t^{12} (\frac{1}{| t |} \frac{d}{d t} [| t |]) + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 3

Combine $\frac{1}{| t |}$ and $t^{12}$ .

$\frac{t^{12}}{| t |} \frac{d}{d t} [| t |] + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 4

The derivative of $| t |$ with respect to $t$ is $\frac{t}{| t |}$ .

$\frac{t^{12}}{| t |} \cdot \frac{t}{| t |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 5

Multiply $\frac{t^{12}}{| t |}$ by $\frac{t}{| t |}$ .

$\frac{t^{12} t}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 6

Multiply $t^{12}$ by $t$ by adding the exponents.

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Step 6.1

Multiply $t^{12}$ by $t$ .

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Step 6.1.1

Raise $t$ to the power of $1$ .

$\frac{t^{12} t^{1}}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 6.1.2

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

$\frac{t^{12 + 1}}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 6.2

Add $12$ and $1$ .

$\frac{t^{13}}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 7

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{t^{13}}{| t \cdot t |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 8

Raise $t$ to the power of $1$ .

$\frac{t^{13}}{| t^{1} t |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 9

Raise $t$ to the power of $1$ .

$\frac{t^{13}}{| t^{1} t^{1} |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 10

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

$\frac{t^{13}}{| t^{1 + 1} |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 11

Add $1$ and $1$ .

$\frac{t^{13}}{| t^{2} |} + \ln (| t |) \frac{d}{d t} [t^{12}]$

Step 12

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 12$ .

$\frac{t^{13}}{| t^{2} |} + \ln (| t |) (12 t^{11})$

Step 13

Simplify.

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Step 13.1

Reorder terms.

$\frac{t^{13}}{| t^{2} |} + 12 t^{11} \ln (| t |)$

Step 13.2

Simplify each term.

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Step 13.2.1

Remove non-negative terms from the absolute value.

$\frac{t^{13}}{t^{2}} + 12 t^{11} \ln (| t |)$

Step 13.2.2

Cancel the common factor of $t^{13}$ and $t^{2}$ .

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Step 13.2.2.1

Factor $t^{2}$ out of $t^{13}$ .

$\frac{t^{2} t^{11}}{t^{2}} + 12 t^{11} \ln (| t |)$

Step 13.2.2.2

Cancel the common factors.

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Step 13.2.2.2.1

Multiply by $1$ .

$\frac{t^{2} t^{11}}{t^{2} \cdot 1} + 12 t^{11} \ln (| t |)$

Step 13.2.2.2.2

Cancel the common factor.

$\frac{t^{2} t^{11}}{t^{2} \cdot 1} + 12 t^{11} \ln (| t |)$

Step 13.2.2.2.3

Rewrite the expression.

$\frac{t^{11}}{1} + 12 t^{11} \ln (| t |)$

Step 13.2.2.2.4

Divide $t^{11}$ by $1$ .

$t^{11} + 12 t^{11} \ln (| t |)$