Calculus Examples

$s = t^{9} \ln (| t |)$

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 $f (t) = t^{9}$ and $g (t) = \ln (| t |)$ .

$t^{9} \frac{d}{d t} [\ln (| t |)] + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$t^{9} (\frac{d}{d u} [\ln (u)] \frac{d}{d t} [| t |]) + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$t^{9} (\frac{1}{u} \frac{d}{d t} [| t |]) + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$t^{9} (\frac{1}{| t |} \frac{d}{d t} [| t |]) + \ln (| t |) \frac{d}{d t} [t^{9}]$

Combine $\frac{1}{| t |}$ and $t^{9}$ .

$\frac{t^{9}}{| t |} \frac{d}{d t} [| t |] + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$\frac{t^{9}}{| t |} \cdot \frac{t}{| t |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

Multiply $\frac{t^{9}}{| t |}$ by $\frac{t}{| t |}$ .

$\frac{t^{9} t}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

Multiply $t^{9}$ by $t$ by adding the exponents.

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Multiply $t^{9}$ by $t$ .

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

$\frac{t^{9} t^{1}}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$\frac{t^{9 + 1}}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

Add $9$ and $1$ .

$\frac{t^{10}}{| t | | t |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$\frac{t^{10}}{| t \cdot t |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$\frac{t^{10}}{| t^{1} t |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$\frac{t^{10}}{| t^{1} t^{1} |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

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

$\frac{t^{10}}{| t^{1 + 1} |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

Add $1$ and $1$ .

$\frac{t^{10}}{| t^{2} |} + \ln (| t |) \frac{d}{d t} [t^{9}]$

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 9$ .

$\frac{t^{10}}{| t^{2} |} + \ln (| t |) (9 t^{8})$

Simplify.

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Reorder terms.

$\frac{t^{10}}{| t^{2} |} + 9 t^{8} \ln (| t |)$

Simplify each term.

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Remove non-negative terms from the absolute value.

$\frac{t^{10}}{t^{2}} + 9 t^{8} \ln (| t |)$

Cancel the common factor of $t^{10}$ and $t^{2}$ .

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Factor $t^{2}$ out of $t^{10}$ .

$\frac{t^{2} t^{8}}{t^{2}} + 9 t^{8} \ln (| t |)$

Cancel the common factors.

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Multiply by $1$ .

$\frac{t^{2} t^{8}}{t^{2} \cdot 1} + 9 t^{8} \ln (| t |)$

Cancel the common factor.

$\frac{t^{2} t^{8}}{t^{2} \cdot 1} + 9 t^{8} \ln (| t |)$

Rewrite the expression.

$\frac{t^{8}}{1} + 9 t^{8} \ln (| t |)$

Divide $t^{8}$ by $1$ .

$t^{8} + 9 t^{8} \ln (| t |)$