Calculus Examples

$\sqrt{t} (1 - t^{2})$

Step 1

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

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

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

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - t^{2}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- t^{2}]$ .

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

Step 3.2

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

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

Step 3.3

Add $0$ and $\frac{d}{d t} [- t^{2}]$ .

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

Step 3.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{2}$ with respect to $t$ is $- \frac{d}{d t} [t^{2}]$ .

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

Step 3.5

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

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

Step 3.6

Multiply $2$ by $- 1$ .

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

Step 4

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

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Step 4.1

Move $t$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Add $2$ and $1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Combine terms.

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Step 14.2.1

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

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

Step 14.2.2

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

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

Step 14.2.3

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

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

Step 14.2.4

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

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Step 14.2.4.1

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

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

Step 14.2.4.2

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

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

Step 14.2.4.3

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

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

Step 14.2.4.4

Combine the numerators over the common denominator.

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

Step 14.2.4.5

Simplify the numerator.

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Step 14.2.4.5.1

Multiply $2$ by $2$ .

$- 2 t^{\frac{3}{2}} + \frac{1}{2 t^{\frac{1}{2}}} - \frac{t^{\frac{4 - 1}{2}}}{2}$

Step 14.2.4.5.2

Subtract $1$ from $4$ .

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

Step 14.2.5

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

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

Step 14.2.6

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

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

Step 14.2.7

Combine the numerators over the common denominator.

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

Step 14.2.8

Multiply $2$ by $- 2$ .

$\frac{- 4 t^{\frac{3}{2}} - t^{\frac{3}{2}}}{2} + \frac{1}{2 t^{\frac{1}{2}}}$

Step 14.2.9

Subtract $t^{\frac{3}{2}}$ from $- 4 t^{\frac{3}{2}}$ .

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

Step 14.2.10

Move the negative in front of the fraction.

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

Step 14.3

Reorder terms.

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