Calculus Examples

$x = (1 + t) \sqrt{t}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (x) = \frac{d}{d t} ((1 + t) t^{\frac{1}{2}})$

Step 3

The derivative of $x$ with respect to $t$ is $x'$ .

$x'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $1$ .

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

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

Add $0$ and $\frac{d}{d t} [t]$ .

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Simplify.

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Step 4.13.1

Apply the distributive property.

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

Step 4.13.2

Combine terms.

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Step 4.13.2.1

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

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

Step 4.13.2.2

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

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

Step 4.13.2.3

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

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

Step 4.13.2.4

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

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Step 4.13.2.4.1

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

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Step 4.13.2.4.1.1

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

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

Step 4.13.2.4.1.2

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

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

Step 4.13.2.4.2

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

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

Step 4.13.2.4.3

Combine the numerators over the common denominator.

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

Step 4.13.2.4.4

Subtract $1$ from $2$ .

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

Step 4.13.2.5

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

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

Step 4.13.2.6

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

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

Step 4.13.2.7

Combine the numerators over the common denominator.

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

Step 4.13.2.8

Move $2$ to the left of $t^{\frac{1}{2}}$ .

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

Step 4.13.2.9

Add $t^{\frac{1}{2}}$ and $2 t^{\frac{1}{2}}$ .

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

Step 5

Reform the equation by setting the left side equal to the right side.

$x' = \frac{1}{2 t^{\frac{1}{2}}} + \frac{3 t^{\frac{1}{2}}}{2}$

Step 6

Replace $x'$ with $\frac{d x}{d t}$ .

$\frac{d x}{d t} = \frac{1}{2 t^{\frac{1}{2}}} + \frac{3 t^{\frac{1}{2}}}{2}$