Calculus Examples

$f (t) = \frac{7 \ln (t)}{t}$

Step 1

Find the first derivative.

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Step 1.1

Since $7$ is constant with respect to $t$ , the derivative of $\frac{7 \ln (t)}{t}$ with respect to $t$ is $7 \frac{d}{d t} [\frac{\ln (t)}{t}]$ .

$7 \frac{d}{d t} [\frac{\ln (t)}{t}]$

Step 1.2

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 $f (t) = \ln (t)$ and $g (t) = t$ .

$7 \frac{t \frac{d}{d t} [\ln (t)] - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 1.3

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

$7 \frac{t \frac{1}{t} - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

$7 \frac{\frac{t}{t} - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 1.4.2

Cancel the common factor of $t$ .

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Step 1.4.2.1

Cancel the common factor.

$7 \frac{\frac{t}{t} - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 1.4.2.2

Rewrite the expression.

$7 \frac{1 - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 1.4.3

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

$7 \frac{1 - \ln (t) \cdot 1}{t^{2}}$

Step 1.4.4

Combine fractions.

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Step 1.4.4.1

Multiply $- 1$ by $1$ .

$7 \frac{1 - \ln (t)}{t^{2}}$

Step 1.4.4.2

Combine $7$ and $\frac{1 - \ln (t)}{t^{2}}$ .

$\frac{7 (1 - \ln (t))}{t^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$\frac{7 \cdot 1 + 7 (- \ln (t))}{t^{2}}$

Step 1.5.2

Simplify each term.

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Step 1.5.2.1

Multiply $7$ by $1$ .

$\frac{7 + 7 (- \ln (t))}{t^{2}}$

Step 1.5.2.2

Multiply $7 (- \ln (t))$ .

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Step 1.5.2.2.1

Multiply $- 1$ by $7$ .

$\frac{7 - 7 \ln (t)}{t^{2}}$

Step 1.5.2.2.2

Simplify $- 7 \ln (t)$ by moving $7$ inside the logarithm.

$f' (t) = \frac{7 - \ln (t^{7})}{t^{2}}$

Step 2

Find the second derivative.

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Step 2.1

$\frac{t^{2} \frac{d}{d t} [7 - \ln (t^{7})] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{{(t^{2})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{t^{2} \frac{d}{d t} [7 - \ln (t^{7})] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{2 \cdot 2}}$

Step 2.2.1.2

Multiply $2$ by $2$ .

$\frac{t^{2} \frac{d}{d t} [7 - \ln (t^{7})] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.2.2

By the Sum Rule, the derivative of $7 - \ln (t^{7})$ with respect to $t$ is $\frac{d}{d t} [7] + \frac{d}{d t} [- \ln (t^{7})]$ .

$\frac{t^{2} (\frac{d}{d t} [7] + \frac{d}{d t} [- \ln (t^{7})]) - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.2.3

Since $7$ is constant with respect to $t$ , the derivative of $7$ with respect to $t$ is $0$ .

$\frac{t^{2} (0 + \frac{d}{d t} [- \ln (t^{7})]) - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.2.4

Add $0$ and $\frac{d}{d t} [- \ln (t^{7})]$ .

$\frac{t^{2} \frac{d}{d t} [- \ln (t^{7})] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.2.5

Since $- 1$ is constant with respect to $t$ , the derivative of $- \ln (t^{7})$ with respect to $t$ is $- \frac{d}{d t} [\ln (t^{7})]$ .

$\frac{t^{2} (- \frac{d}{d t} [\ln (t^{7})]) - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.3

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 $g (t) = t^{7}$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $t^{7}$ .

$\frac{t^{2} (- (\frac{d}{d u} [\ln (u)] \frac{d}{d t} [t^{7}])) - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.3.2

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

$\frac{t^{2} (- (\frac{1}{u} \frac{d}{d t} [t^{7}])) - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.3.3

Replace all occurrences of $u$ with $t^{7}$ .

$\frac{t^{2} (- (\frac{1}{t^{7}} \frac{d}{d t} [t^{7}])) - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4

Differentiate using the Power Rule.

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Step 2.4.1

Combine $t^{2}$ and $\frac{1}{t^{7}}$ .

$\frac{- \frac{t^{2}}{t^{7}} \frac{d}{d t} [t^{7}] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.2

Cancel the common factor of $t^{2}$ and $t^{7}$ .

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Step 2.4.2.1

Multiply by $1$ .

$\frac{- \frac{t^{2} \cdot 1}{t^{7}} \frac{d}{d t} [t^{7}] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.2.2

Cancel the common factors.

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Step 2.4.2.2.1

Factor $t^{2}$ out of $t^{7}$ .

$\frac{- \frac{t^{2} \cdot 1}{t^{2} t^{5}} \frac{d}{d t} [t^{7}] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.2.2.2

Cancel the common factor.

$\frac{- \frac{t^{2} \cdot 1}{t^{2} t^{5}} \frac{d}{d t} [t^{7}] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.2.2.3

Rewrite the expression.

$\frac{- \frac{1}{t^{5}} \frac{d}{d t} [t^{7}] - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 7$ .

$\frac{- \frac{1}{t^{5}} (7 t^{6}) - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4

Simplify terms.

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Step 2.4.4.1

Multiply $7$ by $- 1$ .

$\frac{- 7 \frac{1}{t^{5}} t^{6} - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4.2

Combine $- 7$ and $\frac{1}{t^{5}}$ .

$\frac{\frac{- 7}{t^{5}} t^{6} - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4.3

Combine $\frac{- 7}{t^{5}}$ and $t^{6}$ .

$\frac{\frac{- 7 t^{6}}{t^{5}} - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4.4

Cancel the common factor of $t^{6}$ and $t^{5}$ .

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Step 2.4.4.4.1

Factor $t^{5}$ out of $- 7 t^{6}$ .

$\frac{\frac{t^{5} (- 7 t)}{t^{5}} - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4.4.2

Cancel the common factors.

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Step 2.4.4.4.2.1

Multiply by $1$ .

$\frac{\frac{t^{5} (- 7 t)}{t^{5} \cdot 1} - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4.4.2.2

Cancel the common factor.

$\frac{\frac{t^{5} (- 7 t)}{t^{5} \cdot 1} - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4.4.2.3

Rewrite the expression.

$\frac{\frac{- 7 t}{1} - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.4.4.2.4

Divide $- 7 t$ by $1$ .

$\frac{- 7 t - (7 - \ln (t^{7})) \frac{d}{d t} [t^{2}]}{t^{4}}$

Step 2.4.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$\frac{- 7 t - (7 - \ln (t^{7})) (2 t)}{t^{4}}$

Step 2.4.6

Simplify with factoring out.

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Step 2.4.6.1

Multiply $2$ by $- 1$ .

$\frac{- 7 t - 2 (7 - \ln (t^{7})) t}{t^{4}}$

Step 2.4.6.2

Factor $t$ out of $- 7 t - 2 (7 - \ln (t^{7})) t$ .

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Step 2.4.6.2.1

Factor $t$ out of $- 7 t$ .

$\frac{t \cdot - 7 - 2 (7 - \ln (t^{7})) t}{t^{4}}$

Step 2.4.6.2.2

Factor $t$ out of $- 2 (7 - \ln (t^{7})) t$ .

$\frac{t \cdot - 7 + t (- 2 (7 - \ln (t^{7})))}{t^{4}}$

Step 2.4.6.2.3

Factor $t$ out of $t \cdot - 7 + t (- 2 (7 - \ln (t^{7})))$ .

$\frac{t (- 7 - 2 (7 - \ln (t^{7})))}{t^{4}}$

Step 2.5

Cancel the common factors.

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Step 2.5.1

Factor $t$ out of $t^{4}$ .

$\frac{t (- 7 - 2 (7 - \ln (t^{7})))}{t \cdot t^{3}}$

Step 2.5.2

Cancel the common factor.

$\frac{t (- 7 - 2 (7 - \ln (t^{7})))}{t \cdot t^{3}}$

Step 2.5.3

Rewrite the expression.

$\frac{- 7 - 2 (7 - \ln (t^{7}))}{t^{3}}$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$\frac{- 7 - 2 \cdot 7 - 2 (- \ln (t^{7}))}{t^{3}}$

Step 2.6.2

Simplify the numerator.

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Step 2.6.2.1

Simplify each term.

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Step 2.6.2.1.1

Multiply $- 2$ by $7$ .

$\frac{- 7 - 14 - 2 (- \ln (t^{7}))}{t^{3}}$

Step 2.6.2.1.2

Multiply $- 2 (- \ln (t^{7}))$ .

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Step 2.6.2.1.2.1

Multiply $- 1$ by $- 2$ .

$\frac{- 7 - 14 + 2 \ln (t^{7})}{t^{3}}$

Step 2.6.2.1.2.2

Simplify $2 \ln (t^{7})$ by moving $2$ inside the logarithm.

$\frac{- 7 - 14 + \ln ({(t^{7})}^{2})}{t^{3}}$

Step 2.6.2.1.3

Multiply the exponents in ${(t^{7})}^{2}$ .

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Step 2.6.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- 7 - 14 + \ln (t^{7 \cdot 2})}{t^{3}}$

Step 2.6.2.1.3.2

Multiply $7$ by $2$ .

$\frac{- 7 - 14 + \ln (t^{14})}{t^{3}}$

Step 2.6.2.2

Subtract $14$ from $- 7$ .

$\frac{- 21 + \ln (t^{14})}{t^{3}}$

Step 2.6.3

Rewrite $- 21$ as $- 1 (21)$ .

$\frac{- 1 (21) + \ln (t^{14})}{t^{3}}$

Step 2.6.4

Factor $- 1$ out of $\ln (t^{14})$ .

$\frac{- 1 (21) - 1 (- \ln (t^{14}))}{t^{3}}$

Step 2.6.5

Factor $- 1$ out of $- 1 (21) - 1 (- \ln (t^{14}))$ .

$\frac{- 1 (21 - \ln (t^{14}))}{t^{3}}$

Step 2.6.6

Move the negative in front of the fraction.

$f'' (t) = - \frac{21 - \ln (t^{14})}{t^{3}}$

Step 3

The second derivative of $f (t)$ with respect to $t$ is $- \frac{21 - \ln (t^{14})}{t^{3}}$ .

$- \frac{21 - \ln (t^{14})}{t^{3}}$