Calculus Examples

$\frac{4 t}{3 t^{2} + 27}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $4$ is constant with respect to $t$ , the derivative of $\frac{4 t}{3 t^{2} + 27}$ with respect to $t$ is $4 \frac{d}{d t} [\frac{t}{3 t^{2} + 27}]$ .

$f' (t) = 4 \frac{d}{d t} (\frac{t}{3 t^{2} + 27})$

Step 1.1.2

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 $f (t) = t$ and $g (t) = 3 t^{2} + 27$ .

$f' (t) = 4 (\frac{(3 t^{2} + 27) \frac{d}{d t} (t) - t \frac{d}{d t} (3 t^{2} + 27)}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$f' (t) = 4 (\frac{(3 t^{2} + 27) \cdot 1 - t \frac{d}{d t} (3 t^{2} + 27)}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.2

Multiply $3 t^{2} + 27$ by $1$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - t \frac{d}{d t} (3 t^{2} + 27)}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.3

By the Sum Rule, the derivative of $3 t^{2} + 27$ with respect to $t$ is $\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [27]$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - t (\frac{d}{d t} (3 t^{2}) + \frac{d}{d t} (27))}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.4

Since $3$ is constant with respect to $t$ , the derivative of $3 t^{2}$ with respect to $t$ is $3 \frac{d}{d t} [t^{2}]$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - t (3 \frac{d}{d t} (t^{2}) + \frac{d}{d t} (27))}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.5

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

$f' (t) = 4 (\frac{3 t^{2} + 27 - t (3 (2 t) + \frac{d}{d t} (27))}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.6

Multiply $2$ by $3$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - t (6 t + \frac{d}{d t} (27))}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.7

Since $27$ is constant with respect to $t$ , the derivative of $27$ with respect to $t$ is $0$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - t (6 t + 0)}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.8

Simplify the expression.

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Step 1.1.3.8.1

Add $6 t$ and $0$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - t (6 t)}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.3.8.2

Multiply $6$ by $- 1$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - 6 t \cdot t}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.4

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

$f' (t) = 4 (\frac{3 t^{2} + 27 - 6 (t \cdot t)}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.5

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

$f' (t) = 4 (\frac{3 t^{2} + 27 - 6 (t \cdot t)}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.6

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

$f' (t) = 4 (\frac{3 t^{2} + 27 - 6 t^{1 + 1}}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.7

Add $1$ and $1$ .

$f' (t) = 4 (\frac{3 t^{2} + 27 - 6 t^{2}}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.8

Subtract $6 t^{2}$ from $3 t^{2}$ .

$f' (t) = 4 (\frac{- 3 t^{2} + 27}{{(3 t^{2} + 27)}^{2}})$

Step 1.1.9

Combine $4$ and $\frac{- 3 t^{2} + 27}{{(3 t^{2} + 27)}^{2}}$ .

$f' (t) = \frac{4 (- 3 t^{2} + 27)}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10

Simplify.

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Step 1.1.10.1

Apply the distributive property.

$f' (t) = \frac{4 (- 3 t^{2}) + 4 \cdot 27}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.2

Simplify each term.

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Step 1.1.10.2.1

Multiply $- 3$ by $4$ .

$f' (t) = \frac{- 12 t^{2} + 4 \cdot 27}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.2.2

Multiply $4$ by $27$ .

$f' (t) = \frac{- 12 t^{2} + 108}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.3

Simplify the numerator.

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Step 1.1.10.3.1

Factor $12$ out of $- 12 t^{2} + 108$ .

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Step 1.1.10.3.1.1

Factor $12$ out of $- 12 t^{2}$ .

$f' (t) = \frac{12 (- t^{2}) + 108}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.3.1.2

Factor $12$ out of $108$ .

$f' (t) = \frac{12 (- t^{2}) + 12 (9)}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.3.1.3

Factor $12$ out of $12 (- t^{2}) + 12 (9)$ .

$f' (t) = \frac{12 (- t^{2} + 9)}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.3.2

Rewrite $9$ as $3^{2}$ .

$f' (t) = \frac{12 (- t^{2} + 3^{2})}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.3.3

Reorder $- t^{2}$ and $3^{2}$ .

$f' (t) = \frac{12 (3^{2} - t^{2})}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = t$ .

$f' (t) = \frac{12 (3 + t) (3 - t)}{{(3 t^{2} + 27)}^{2}}$

Step 1.1.10.4

Simplify the denominator.

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Step 1.1.10.4.1

Factor $3$ out of $3 t^{2} + 27$ .

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Step 1.1.10.4.1.1

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

$f' (t) = \frac{12 (3 + t) (3 - t)}{{(3 (t^{2}) + 27)}^{2}}$

Step 1.1.10.4.1.2

Factor $3$ out of $27$ .

$f' (t) = \frac{12 (3 + t) (3 - t)}{{(3 t^{2} + 3 \cdot 9)}^{2}}$

Step 1.1.10.4.1.3

Factor $3$ out of $3 t^{2} + 3 \cdot 9$ .

$f' (t) = \frac{12 (3 + t) (3 - t)}{{(3 (t^{2} + 9))}^{2}}$

Step 1.1.10.4.2

Apply the product rule to $3 (t^{2} + 9)$ .

$f' (t) = \frac{12 (3 + t) (3 - t)}{3^{2} {(t^{2} + 9)}^{2}}$

Step 1.1.10.4.3

Raise $3$ to the power of $2$ .

$f' (t) = \frac{12 (3 + t) (3 - t)}{9 {(t^{2} + 9)}^{2}}$

Step 1.1.10.5

Cancel the common factor of $12$ and $9$ .

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Step 1.1.10.5.1

Factor $3$ out of $12 (3 + t) (3 - t)$ .

$f' (t) = \frac{3 (4 (3 + t) (3 - t))}{9 {(t^{2} + 9)}^{2}}$

Step 1.1.10.5.2

Cancel the common factors.

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Step 1.1.10.5.2.1

Factor $3$ out of $9 {(t^{2} + 9)}^{2}$ .

$f' (t) = \frac{3 (4 (3 + t) (3 - t))}{3 (3 {(t^{2} + 9)}^{2})}$

Step 1.1.10.5.2.2

Cancel the common factor.

$f' (t) = \frac{3 (4 (3 + t) (3 - t))}{3 (3 {(t^{2} + 9)}^{2})}$

Step 1.1.10.5.2.3

Rewrite the expression.

$f' (t) = \frac{4 (3 + t) (3 - t)}{3 {(t^{2} + 9)}^{2}}$

Step 1.2

The first derivative of $f (t)$ with respect to $t$ is $\frac{4 (3 + t) (3 - t)}{3 {(t^{2} + 9)}^{2}}$ .

$\frac{4 (3 + t) (3 - t)}{3 {(t^{2} + 9)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{4 (3 + t) (3 - t)}{3 {(t^{2} + 9)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{4 (3 + t) (3 - t)}{3 {(t^{2} + 9)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 (3 + t) (3 - t) = 0$

Step 2.3

Solve the equation for $t$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 + t = 0$

$3 - t = 0$

Step 2.3.2

Set $3 + t$ equal to $0$ and solve for $t$ .

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Step 2.3.2.1

Set $3 + t$ equal to $0$ .

$3 + t = 0$

Step 2.3.2.2

Subtract $3$ from both sides of the equation.

$t = - 3$

Step 2.3.3

Set $3 - t$ equal to $0$ and solve for $t$ .

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Step 2.3.3.1

Set $3 - t$ equal to $0$ .

$3 - t = 0$

Step 2.3.3.2

Solve $3 - t = 0$ for $t$ .

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Step 2.3.3.2.1

Subtract $3$ from both sides of the equation.

$- t = - 3$

Step 2.3.3.2.2

Divide each term in $- t = - 3$ by $- 1$ and simplify.

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Step 2.3.3.2.2.1

Divide each term in $- t = - 3$ by $- 1$ .

$\frac{- t}{- 1} = \frac{- 3}{- 1}$

Step 2.3.3.2.2.2

Simplify the left side.

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Step 2.3.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{- 3}{- 1}$

Step 2.3.3.2.2.2.2

Divide $t$ by $1$ .

$t = \frac{- 3}{- 1}$

Step 2.3.3.2.2.3

Simplify the right side.

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Step 2.3.3.2.2.3.1

Divide $- 3$ by $- 1$ .

$t = 3$

Step 2.3.4

The final solution is all the values that make $4 (3 + t) (3 - t) = 0$ true.

$t = - 3, 3$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{4 t}{3 t^{2} + 27}$ at each $t$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $t = - 3$ .

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Step 4.1.1

Substitute $- 3$ for $t$ .

$\frac{4 (- 3)}{3 {(- 3)}^{2} + 27}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $- 3$ and $3 {(- 3)}^{2} + 27$ .

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Step 4.1.2.1.1

Factor $3$ out of $4 (- 3)$ .

$\frac{3 (4 \cdot (- 1))}{3 {(- 3)}^{2} + 27}$

Step 4.1.2.1.2

Cancel the common factors.

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Step 4.1.2.1.2.1

Factor $3$ out of $3 {(- 3)}^{2}$ .

$\frac{3 (4 \cdot (- 1))}{3 ({(- 3)}^{2}) + 27}$

Step 4.1.2.1.2.2

Factor $3$ out of $27$ .

$\frac{3 (4 \cdot (- 1))}{3 {(- 3)}^{2} + 3 \cdot 9}$

Step 4.1.2.1.2.3

Factor $3$ out of $3 {(- 3)}^{2} + 3 \cdot 9$ .

$\frac{3 (4 \cdot (- 1))}{3 ({(- 3)}^{2} + 9)}$

Step 4.1.2.1.2.4

Cancel the common factor.

$\frac{3 (4 \cdot (- 1))}{3 ({(- 3)}^{2} + 9)}$

Step 4.1.2.1.2.5

Rewrite the expression.

$\frac{4 \cdot (- 1)}{{(- 3)}^{2} + 9}$

Step 4.1.2.2

Multiply $4$ by $- 1$ .

$\frac{- 4}{{(- 3)}^{2} + 9}$

Step 4.1.2.3

Simplify the denominator.

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Step 4.1.2.3.1

Raise $- 3$ to the power of $2$ .

$\frac{- 4}{9 + 9}$

Step 4.1.2.3.2

Add $9$ and $9$ .

$\frac{- 4}{18}$

Step 4.1.2.4

Reduce the expression by cancelling the common factors.

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Step 4.1.2.4.1

Cancel the common factor of $- 4$ and $18$ .

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Step 4.1.2.4.1.1

Factor $2$ out of $- 4$ .

$\frac{2 (- 2)}{18}$

Step 4.1.2.4.1.2

Cancel the common factors.

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Step 4.1.2.4.1.2.1

Factor $2$ out of $18$ .

$\frac{2 \cdot - 2}{2 \cdot 9}$

Step 4.1.2.4.1.2.2

Cancel the common factor.

$\frac{2 \cdot - 2}{2 \cdot 9}$

Step 4.1.2.4.1.2.3

Rewrite the expression.

$\frac{- 2}{9}$

Step 4.1.2.4.2

Move the negative in front of the fraction.

$- \frac{2}{9}$

Step 4.2

Evaluate at $t = 3$ .

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Step 4.2.1

Substitute $3$ for $t$ .

$\frac{4 (3)}{3 {(3)}^{2} + 27}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $3$ and $3 {(3)}^{2} + 27$ .

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Step 4.2.2.1.1

Factor $3$ out of $4 (3)$ .

$\frac{3 \cdot 4}{3 {(3)}^{2} + 27}$

Step 4.2.2.1.2

Cancel the common factors.

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Step 4.2.2.1.2.1

Factor $3$ out of $3 {(3)}^{2}$ .

$\frac{3 \cdot 4}{3 ({(3)}^{2}) + 27}$

Step 4.2.2.1.2.2

Factor $3$ out of $27$ .

$\frac{3 \cdot 4}{3 {(3)}^{2} + 3 \cdot 9}$

Step 4.2.2.1.2.3

Factor $3$ out of $3 {(3)}^{2} + 3 \cdot 9$ .

$\frac{3 \cdot 4}{3 ({(3)}^{2} + 9)}$

Step 4.2.2.1.2.4

Cancel the common factor.

$\frac{3 \cdot 4}{3 ({(3)}^{2} + 9)}$

Step 4.2.2.1.2.5

Rewrite the expression.

$\frac{4}{{(3)}^{2} + 9}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Raise $3$ to the power of $2$ .

$\frac{4}{9 + 9}$

Step 4.2.2.2.2

Add $9$ and $9$ .

$\frac{4}{18}$

Step 4.2.2.3

Cancel the common factor of $4$ and $18$ .

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Step 4.2.2.3.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{18}$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 2}{2 \cdot 9}$

Step 4.2.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 9}$

Step 4.2.2.3.2.3

Rewrite the expression.

$\frac{2}{9}$

Step 4.3

List all of the points.

$(- 3, - \frac{2}{9}), (3, \frac{2}{9})$

Step 5