Calculus Examples

$\frac{d^{2}}{d t^{2}} (t^{12} \ln (t))$

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

Find the first derivative.

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Step 2.1

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 2.2

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

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $t^{12}$ and $t$ .

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Step 2.3.2.1

Factor $t$ out of $t^{12}$ .

$\frac{t \cdot t^{11}}{t} + \ln (t) \frac{d}{d t} [t^{12}]$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

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

$\frac{t \cdot t^{11}}{t^{1}} + \ln (t) \frac{d}{d t} [t^{12}]$

Step 2.3.2.2.2

Factor $t$ out of $t^{1}$ .

$\frac{t \cdot t^{11}}{t \cdot 1} + \ln (t) \frac{d}{d t} [t^{12}]$

Step 2.3.2.2.3

Cancel the common factor.

$\frac{t \cdot t^{11}}{t \cdot 1} + \ln (t) \frac{d}{d t} [t^{12}]$

Step 2.3.2.2.4

Rewrite the expression.

$\frac{t^{11}}{1} + \ln (t) \frac{d}{d t} [t^{12}]$

Step 2.3.2.2.5

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

$t^{11} + \ln (t) \frac{d}{d t} [t^{12}]$

Step 2.3.3

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

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

Step 2.3.4

Reorder terms.

$f' (t) = t^{11} + 12 t^{11} \ln (t)$

Step 3

Find the second derivative.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $t^{11} + 12 t^{11} \ln (t)$ with respect to $t$ is $\frac{d}{d t} [t^{11}] + \frac{d}{d t} [12 t^{11} \ln (t)]$ .

$\frac{d}{d t} [t^{11}] + \frac{d}{d t} [12 t^{11} \ln (t)]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 11$ .

$11 t^{10} + \frac{d}{d t} [12 t^{11} \ln (t)]$

Step 3.2

Evaluate $\frac{d}{d t} [12 t^{11} \ln (t)]$ .

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Step 3.2.1

Since $12$ is constant with respect to $t$ , the derivative of $12 t^{11} \ln (t)$ with respect to $t$ is $12 \frac{d}{d t} [t^{11} \ln (t)]$ .

$11 t^{10} + 12 \frac{d}{d t} [t^{11} \ln (t)]$

Step 3.2.2

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 $f (t) = t^{11}$ and $g (t) = \ln (t)$ .

$11 t^{10} + 12 (t^{11} \frac{d}{d t} [\ln (t)] + \ln (t) \frac{d}{d t} [t^{11}])$

Step 3.2.3

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

$11 t^{10} + 12 (t^{11} \frac{1}{t} + \ln (t) \frac{d}{d t} [t^{11}])$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 11$ .

$11 t^{10} + 12 (t^{11} \frac{1}{t} + \ln (t) (11 t^{10}))$

Step 3.2.5

Combine $t^{11}$ and $\frac{1}{t}$ .

$11 t^{10} + 12 (\frac{t^{11}}{t} + \ln (t) (11 t^{10}))$

Step 3.2.6

Cancel the common factor of $t^{11}$ and $t$ .

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Step 3.2.6.1

Factor $t$ out of $t^{11}$ .

$11 t^{10} + 12 (\frac{t \cdot t^{10}}{t} + \ln (t) (11 t^{10}))$

Step 3.2.6.2

Cancel the common factors.

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Step 3.2.6.2.1

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

$11 t^{10} + 12 (\frac{t \cdot t^{10}}{t^{1}} + \ln (t) (11 t^{10}))$

Step 3.2.6.2.2

Factor $t$ out of $t^{1}$ .

$11 t^{10} + 12 (\frac{t \cdot t^{10}}{t \cdot 1} + \ln (t) (11 t^{10}))$

Step 3.2.6.2.3

Cancel the common factor.

$11 t^{10} + 12 (\frac{t \cdot t^{10}}{t \cdot 1} + \ln (t) (11 t^{10}))$

Step 3.2.6.2.4

Rewrite the expression.

$11 t^{10} + 12 (\frac{t^{10}}{1} + \ln (t) (11 t^{10}))$

Step 3.2.6.2.5

Divide $t^{10}$ by $1$ .

$11 t^{10} + 12 (t^{10} + \ln (t) (11 t^{10}))$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$11 t^{10} + 12 t^{10} + 12 (\ln (t) (11 t^{10}))$

Step 3.3.2

Combine terms.

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Step 3.3.2.1

Multiply $11$ by $12$ .

$11 t^{10} + 12 t^{10} + 132 (\ln (t) (t^{10}))$

Step 3.3.2.2

Add $11 t^{10}$ and $12 t^{10}$ .

$23 t^{10} + 132 \ln (t) t^{10}$

Step 3.3.3

Reorder terms.

$f'' (t) = 23 t^{10} + 132 t^{10} \ln (t)$