Calculus Examples

$\frac{t^{3} + 2 t^{2}}{\sqrt{t}}$

Step 1

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

$\frac{d}{d t} [\frac{t^{3} + 2 t^{2}}{t^{\frac{1}{2}}}]$

Step 2

Factor $t^{2}$ out of $t^{3} + 2 t^{2}$ .

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Step 2.1

Factor $t^{2}$ out of $t^{3}$ .

$\frac{d}{d t} [\frac{t^{2} t + 2 t^{2}}{t^{\frac{1}{2}}}]$

Step 2.2

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

$\frac{d}{d t} [\frac{t^{2} t + t^{2} \cdot 2}{t^{\frac{1}{2}}}]$

Step 2.3

Factor $t^{2}$ out of $t^{2} t + t^{2} \cdot 2$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $2$ .

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

Step 4.6.2

Add $- 1$ and $4$ .

$\frac{d}{d t} [t^{\frac{3}{2}} (t + 2)]$

Step 5

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 $f (t) = t^{\frac{3}{2}}$ and $g (t) = t + 2$ .

$t^{\frac{3}{2}} \frac{d}{d t} [t + 2] + (t + 2) \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $t + 2$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [2]$ .

$t^{\frac{3}{2}} (\frac{d}{d t} [t] + \frac{d}{d t} [2]) + (t + 2) \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 6.2

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

$t^{\frac{3}{2}} (1 + \frac{d}{d t} [2]) + (t + 2) \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 6.3

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$t^{\frac{3}{2}} (1 + 0) + (t + 2) \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 6.4

Simplify the expression.

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Step 6.4.1

Add $1$ and $0$ .

$t^{\frac{3}{2}} \cdot 1 + (t + 2) \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 6.4.2

Multiply $t^{\frac{3}{2}}$ by $1$ .

$t^{\frac{3}{2}} + (t + 2) \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 6.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{3}{2}$ .

$t^{\frac{3}{2}} + (t + 2) (\frac{3}{2} t^{\frac{3}{2} - 1})$

Step 7

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

$t^{\frac{3}{2}} + (t + 2) (\frac{3}{2} t^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 8

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

$t^{\frac{3}{2}} + (t + 2) (\frac{3}{2} t^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 9

Combine the numerators over the common denominator.

$t^{\frac{3}{2}} + (t + 2) (\frac{3}{2} t^{\frac{3 - 1 \cdot 2}{2}})$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $3$ .

$t^{\frac{3}{2}} + (t + 2) (\frac{3}{2} t^{\frac{1}{2}})$

Step 11

Combine $\frac{3}{2}$ and $t^{\frac{1}{2}}$ .

$t^{\frac{3}{2}} + (t + 2) \frac{3 t^{\frac{1}{2}}}{2}$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$t^{\frac{3}{2}} + t \frac{3 t^{\frac{1}{2}}}{2} + 2 \frac{3 t^{\frac{1}{2}}}{2}$

Step 12.2

Combine terms.

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Step 12.2.1

Combine $t$ and $\frac{3 t^{\frac{1}{2}}}{2}$ .

$t^{\frac{3}{2}} + \frac{t (3 t^{\frac{1}{2}})}{2} + 2 \frac{3 t^{\frac{1}{2}}}{2}$

Step 12.2.2

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

$t^{\frac{3}{2}} + \frac{3 (t^{1} t^{\frac{1}{2}})}{2} + 2 \frac{3 t^{\frac{1}{2}}}{2}$

Step 12.2.3

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

$t^{\frac{3}{2}} + \frac{3 t^{1 + \frac{1}{2}}}{2} + 2 \frac{3 t^{\frac{1}{2}}}{2}$

Step 12.2.4

Write $1$ as a fraction with a common denominator.

$t^{\frac{3}{2}} + \frac{3 t^{\frac{2}{2} + \frac{1}{2}}}{2} + 2 \frac{3 t^{\frac{1}{2}}}{2}$

Step 12.2.5

Combine the numerators over the common denominator.

$t^{\frac{3}{2}} + \frac{3 t^{\frac{2 + 1}{2}}}{2} + 2 \frac{3 t^{\frac{1}{2}}}{2}$

Step 12.2.6

Add $2$ and $1$ .

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + 2 \frac{3 t^{\frac{1}{2}}}{2}$

Step 12.2.7

Combine $2$ and $\frac{3 t^{\frac{1}{2}}}{2}$ .

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + \frac{2 (3 t^{\frac{1}{2}})}{2}$

Step 12.2.8

Multiply $3$ by $2$ .

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + \frac{6 t^{\frac{1}{2}}}{2}$

Step 12.2.9

Factor $2$ out of $6 t^{\frac{1}{2}}$ .

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + \frac{2 (3 t^{\frac{1}{2}})}{2}$

Step 12.2.10

Cancel the common factors.

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Step 12.2.10.1

Factor $2$ out of $2$ .

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + \frac{2 (3 t^{\frac{1}{2}})}{2 (1)}$

Step 12.2.10.2

Cancel the common factor.

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + \frac{2 (3 t^{\frac{1}{2}})}{2 \cdot 1}$

Step 12.2.10.3

Rewrite the expression.

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + \frac{3 t^{\frac{1}{2}}}{1}$

Step 12.2.10.4

Divide $3 t^{\frac{1}{2}}$ by $1$ .

$t^{\frac{3}{2}} + \frac{3 t^{\frac{3}{2}}}{2} + 3 t^{\frac{1}{2}}$

Step 12.2.11

To write $t^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$t^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{3 t^{\frac{3}{2}}}{2} + 3 t^{\frac{1}{2}}$

Step 12.2.12

Combine $t^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$\frac{t^{\frac{3}{2}} \cdot 2}{2} + \frac{3 t^{\frac{3}{2}}}{2} + 3 t^{\frac{1}{2}}$

Step 12.2.13

Combine the numerators over the common denominator.

$\frac{t^{\frac{3}{2}} \cdot 2 + 3 t^{\frac{3}{2}}}{2} + 3 t^{\frac{1}{2}}$

Step 12.2.14

Move $2$ to the left of $t^{\frac{3}{2}}$ .

$\frac{2 \cdot t^{\frac{3}{2}} + 3 t^{\frac{3}{2}}}{2} + 3 t^{\frac{1}{2}}$

Step 12.2.15

Add $2 t^{\frac{3}{2}}$ and $3 t^{\frac{3}{2}}$ .

$\frac{5 t^{\frac{3}{2}}}{2} + 3 t^{\frac{1}{2}}$

Step 12.3

Reorder terms.

$3 t^{\frac{1}{2}} + \frac{5 t^{\frac{3}{2}}}{2}$