Calculus Examples

$(0.1 t + 1) \ln (\sqrt{t})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{d}{d t} [(0.1 t + 1) \ln (t^{\frac{1}{2}})]$

Step 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 $f (t) = 0.1 t + 1$ and $g (t) = \ln (t^{\frac{1}{2}})$ .

$(0.1 t + 1) \frac{d}{d t} [\ln (t^{\frac{1}{2}})] + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 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 $g (t) = t^{\frac{1}{2}}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $t^{\frac{1}{2}}$ .

$(0.1 t + 1) (\frac{d}{d u} [\ln (u)] \frac{d}{d t} [t^{\frac{1}{2}}]) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 3.2

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

$(0.1 t + 1) (\frac{1}{u} \frac{d}{d t} [t^{\frac{1}{2}}]) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 3.3

Replace all occurrences of $u$ with $t^{\frac{1}{2}}$ .

$(0.1 t + 1) (\frac{1}{t^{\frac{1}{2}}} \frac{d}{d t} [t^{\frac{1}{2}}]) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 4

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

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2} t^{\frac{1}{2} - 1}) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 6

Combine $- 1$ and $\frac{2}{2}$ .

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 7

Combine the numerators over the common denominator.

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}}) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2} t^{\frac{1 - 2}{2}}) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 8.2

Subtract $2$ from $1$ .

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2} t^{\frac{- 1}{2}}) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 9

Move the negative in front of the fraction.

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} (\frac{1}{2} t^{- \frac{1}{2}}) + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 10

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

$(0.1 t + 1) \frac{1}{t^{\frac{1}{2}}} \frac{t^{- \frac{1}{2}}}{2} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 11

Multiply $\frac{t^{- \frac{1}{2}}}{2}$ by $\frac{1}{t^{\frac{1}{2}}}$ .

$(0.1 t + 1) \frac{t^{- \frac{1}{2}}}{2 t^{\frac{1}{2}}} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 12

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(0.1 t + 1) \frac{1}{2 t^{\frac{1}{2}} t^{\frac{1}{2}}} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 13

Simplify the denominator.

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Step 13.1

Multiply $t^{\frac{1}{2}}$ by $t^{\frac{1}{2}}$ by adding the exponents.

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Step 13.1.1

Move $t^{\frac{1}{2}}$ .

$(0.1 t + 1) \frac{1}{2 (t^{\frac{1}{2}} t^{\frac{1}{2}})} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 13.1.2

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

$(0.1 t + 1) \frac{1}{2 t^{\frac{1}{2} + \frac{1}{2}}} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 13.1.3

Combine the numerators over the common denominator.

$(0.1 t + 1) \frac{1}{2 t^{\frac{1 + 1}{2}}} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 13.1.4

Add $1$ and $1$ .

$(0.1 t + 1) \frac{1}{2 t^{\frac{2}{2}}} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 13.1.5

Divide $2$ by $2$ .

$(0.1 t + 1) \frac{1}{2 t^{1}} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 13.2

Simplify $2 t^{1}$ .

$(0.1 t + 1) \frac{1}{2 t} + \ln (t^{\frac{1}{2}}) \frac{d}{d t} [0.1 t + 1]$

Step 14

By the Sum Rule, the derivative of $0.1 t + 1$ with respect to $t$ is $\frac{d}{d t} [0.1 t] + \frac{d}{d t} [1]$ .

$(0.1 t + 1) \frac{1}{2 t} + \ln (t^{\frac{1}{2}}) (\frac{d}{d t} [0.1 t] + \frac{d}{d t} [1])$

Step 15

Since $0.1$ is constant with respect to $t$ , the derivative of $0.1 t$ with respect to $t$ is $0.1 \frac{d}{d t} [t]$ .

$(0.1 t + 1) \frac{1}{2 t} + \ln (t^{\frac{1}{2}}) (0.1 \frac{d}{d t} [t] + \frac{d}{d t} [1])$

Step 16

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

$(0.1 t + 1) \frac{1}{2 t} + \ln (t^{\frac{1}{2}}) (0.1 \cdot 1 + \frac{d}{d t} [1])$

Step 17

Multiply $0.1$ by $1$ .

$(0.1 t + 1) \frac{1}{2 t} + \ln (t^{\frac{1}{2}}) (0.1 + \frac{d}{d t} [1])$

Step 18

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$(0.1 t + 1) \frac{1}{2 t} + \ln (t^{\frac{1}{2}}) (0.1 + 0)$

Step 19

Simplify the expression.

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Step 19.1

Add $0.1$ and $0$ .

$(0.1 t + 1) \frac{1}{2 t} + \ln (t^{\frac{1}{2}}) \cdot 0.1$

Step 19.2

Move $0.1$ to the left of $\ln (t^{\frac{1}{2}})$ .

$(0.1 t + 1) \frac{1}{2 t} + 0.1 \cdot \ln (t^{\frac{1}{2}})$

Step 20

Simplify.

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Step 20.1

Apply the distributive property.

$0.1 t \frac{1}{2 t} + 1 \frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}})$

Step 20.2

Combine terms.

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Step 20.2.1

Combine $0.1$ and $\frac{1}{2 t}$ .

$t \frac{0.1}{2 t} + 1 \frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}})$

Step 20.2.2

Combine $t$ and $\frac{0.1}{2 t}$ .

$\frac{t \cdot 0.1}{2 t} + 1 \frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}})$

Step 20.2.3

Move $0.1$ to the left of $t$ .

$\frac{0.1 \cdot t}{2 t} + 1 \frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}})$

Step 20.2.4

Cancel the common factor of $t$ .

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Step 20.2.4.1

Cancel the common factor.

$\frac{0.1 t}{2 t} + 1 \frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}})$

Step 20.2.4.2

Rewrite the expression.

$\frac{0.1}{2} + 1 \frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}})$

Step 20.2.5

Multiply $\frac{1}{2 t}$ by $1$ .

$\frac{0.1}{2} + \frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}})$

Step 20.3

Reorder terms.

$\frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}}) + \frac{0.1}{2}$

Step 20.4

Divide $0.1$ by $2$ .

$\frac{1}{2 t} + 0.1 \ln (t^{\frac{1}{2}}) + 0.05$