Calculus Examples

$g (t) = {(4 t + 3 t)}^{2} {(3 t^{2} - 9)}^{- 3}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Add $4 t$ and $3 t$ .

$\frac{d}{d t} [{(7 t)}^{2} {(3 t^{2} - 9)}^{- 3}]$

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $7 t$ .

$\frac{d}{d t} [7^{2} t^{2} {(3 t^{2} - 9)}^{- 3}]$

Step 1.2.2

Raise $7$ to the power of $2$ .

$\frac{d}{d t} [49 t^{2} {(3 t^{2} - 9)}^{- 3}]$

Step 1.3

Since $49$ is constant with respect to $t$ , the derivative of $49 t^{2} {(3 t^{2} - 9)}^{- 3}$ with respect to $t$ is $49 \frac{d}{d t} [t^{2} {(3 t^{2} - 9)}^{- 3}]$ .

$49 \frac{d}{d t} [t^{2} {(3 t^{2} - 9)}^{- 3}]$

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 $f (t) = t^{2}$ and $g (t) = {(3 t^{2} - 9)}^{- 3}$ .

$49 (t^{2} \frac{d}{d t} [{(3 t^{2} - 9)}^{- 3}] + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 3}$ and $g (t) = 3 t^{2} - 9$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $3 t^{2} - 9$ .

$49 (t^{2} (\frac{d}{d u} [u^{- 3}] \frac{d}{d t} [3 t^{2} - 9]) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 3.2

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

$49 (t^{2} (- 3 u^{- 4} \frac{d}{d t} [3 t^{2} - 9]) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 3.3

Replace all occurrences of $u$ with $3 t^{2} - 9$ .

$49 (t^{2} (- 3 {(3 t^{2} - 9)}^{- 4} \frac{d}{d t} [3 t^{2} - 9]) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $3 t^{2} - 9$ with respect to $t$ is $\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [- 9]$ .

$49 (t^{2} (- 3 {(3 t^{2} - 9)}^{- 4} (\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [- 9])) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 4.2

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

$49 (t^{2} (- 3 {(3 t^{2} - 9)}^{- 4} (3 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 9])) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 4.3

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

$49 (t^{2} (- 3 {(3 t^{2} - 9)}^{- 4} (3 (2 t) + \frac{d}{d t} [- 9])) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 4.4

Multiply $2$ by $3$ .

$49 (t^{2} (- 3 {(3 t^{2} - 9)}^{- 4} (6 t + \frac{d}{d t} [- 9])) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 4.5

Since $- 9$ is constant with respect to $t$ , the derivative of $- 9$ with respect to $t$ is $0$ .

$49 (t^{2} (- 3 {(3 t^{2} - 9)}^{- 4} (6 t + 0)) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $6 t$ and $0$ .

$49 (t^{2} (- 3 {(3 t^{2} - 9)}^{- 4} (6 t)) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 4.6.2

Multiply $6$ by $- 3$ .

$49 (t^{2} (- 18 {(3 t^{2} - 9)}^{- 4} t) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 5

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

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Step 5.1

Move $t$ .

$49 (t \cdot t^{2} (- 18 {(3 t^{2} - 9)}^{- 4}) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 5.2

Multiply $t$ by $t^{2}$ .

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Step 5.2.1

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

$49 (t^{1} t^{2} (- 18 {(3 t^{2} - 9)}^{- 4}) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 5.2.2

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

$49 (t^{1 + 2} (- 18 {(3 t^{2} - 9)}^{- 4}) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 5.3

Add $1$ and $2$ .

$49 (t^{3} (- 18 {(3 t^{2} - 9)}^{- 4}) + {(3 t^{2} - 9)}^{- 3} \frac{d}{d t} [t^{2}])$

Step 6

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

$49 (t^{3} (- 18 {(3 t^{2} - 9)}^{- 4}) + {(3 t^{2} - 9)}^{- 3} (2 t))$

Step 7

Move $2$ to the left of ${(3 t^{2} - 9)}^{- 3}$ .

$49 (t^{3} (- 18 {(3 t^{2} - 9)}^{- 4}) + 2 \cdot {(3 t^{2} - 9)}^{- 3} t)$

Step 8

Simplify.

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Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$49 (t^{3} (- 18 \frac{1}{{(3 t^{2} - 9)}^{4}}) + 2 {(3 t^{2} - 9)}^{- 3} t)$

Step 8.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$49 (t^{3} (- 18 \frac{1}{{(3 t^{2} - 9)}^{4}}) + 2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.3

Apply the distributive property.

$49 (t^{3} (- 18 \frac{1}{{(3 t^{2} - 9)}^{4}})) + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4

Combine terms.

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Step 8.4.1

Combine $- 18$ and $\frac{1}{{(3 t^{2} - 9)}^{4}}$ .

$49 (t^{3} \frac{- 18}{{(3 t^{2} - 9)}^{4}}) + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.2

Move the negative in front of the fraction.

$49 (t^{3} (- \frac{18}{{(3 t^{2} - 9)}^{4}})) + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.3

Combine $t^{3}$ and $\frac{18}{{(3 t^{2} - 9)}^{4}}$ .

$49 (- \frac{t^{3} \cdot 18}{{(3 t^{2} - 9)}^{4}}) + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.4

Move $18$ to the left of $t^{3}$ .

$49 (- \frac{18 \cdot t^{3}}{{(3 t^{2} - 9)}^{4}}) + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.5

Multiply $- 1$ by $49$ .

$- 49 \frac{18 t^{3}}{{(3 t^{2} - 9)}^{4}} + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.6

Combine $- 49$ and $\frac{18 t^{3}}{{(3 t^{2} - 9)}^{4}}$ .

$\frac{- 49 (18 t^{3})}{{(3 t^{2} - 9)}^{4}} + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.7

Multiply $18$ by $- 49$ .

$\frac{- 882 t^{3}}{{(3 t^{2} - 9)}^{4}} + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.8

Move the negative in front of the fraction.

$- \frac{882 t^{3}}{{(3 t^{2} - 9)}^{4}} + 49 (2 \frac{1}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.9

Combine $2$ and $\frac{1}{{(3 t^{2} - 9)}^{3}}$ .

$- \frac{882 t^{3}}{{(3 t^{2} - 9)}^{4}} + 49 (\frac{2}{{(3 t^{2} - 9)}^{3}} t)$

Step 8.4.10

Combine $\frac{2}{{(3 t^{2} - 9)}^{3}}$ and $t$ .

$- \frac{882 t^{3}}{{(3 t^{2} - 9)}^{4}} + 49 \frac{2 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.4.11

Combine $49$ and $\frac{2 t}{{(3 t^{2} - 9)}^{3}}$ .

$- \frac{882 t^{3}}{{(3 t^{2} - 9)}^{4}} + \frac{49 (2 t)}{{(3 t^{2} - 9)}^{3}}$

Step 8.4.12

Multiply $2$ by $49$ .

$- \frac{882 t^{3}}{{(3 t^{2} - 9)}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5

Simplify each term.

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Step 8.5.1

Simplify the denominator.

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Step 8.5.1.1

Factor $3$ out of $3 t^{2} - 9$ .

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Step 8.5.1.1.1

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

$- \frac{882 t^{3}}{{(3 (t^{2}) - 9)}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.1.1.2

Factor $3$ out of $- 9$ .

$- \frac{882 t^{3}}{{(3 t^{2} + 3 \cdot - 3)}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.1.1.3

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

$- \frac{882 t^{3}}{{(3 (t^{2} - 3))}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.1.2

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

$- \frac{882 t^{3}}{3^{4} {(t^{2} - 3)}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.1.3

Raise $3$ to the power of $4$ .

$- \frac{882 t^{3}}{81 {(t^{2} - 3)}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.2

Cancel the common factor of $882$ and $81$ .

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Step 8.5.2.1

Factor $9$ out of $882 t^{3}$ .

$- \frac{9 (98 t^{3})}{81 {(t^{2} - 3)}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.2.2

Cancel the common factors.

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Step 8.5.2.2.1

Factor $9$ out of $81 {(t^{2} - 3)}^{4}$ .

$- \frac{9 (98 t^{3})}{9 (9 {(t^{2} - 3)}^{4})} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.2.2.2

Cancel the common factor.

$- \frac{9 (98 t^{3})}{9 (9 {(t^{2} - 3)}^{4})} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.2.2.3

Rewrite the expression.

$- \frac{98 t^{3}}{9 {(t^{2} - 3)}^{4}} + \frac{98 t}{{(3 t^{2} - 9)}^{3}}$

Step 8.5.3

Simplify the denominator.

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Step 8.5.3.1

Factor $3$ out of $3 t^{2} - 9$ .

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Step 8.5.3.1.1

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

$- \frac{98 t^{3}}{9 {(t^{2} - 3)}^{4}} + \frac{98 t}{{(3 (t^{2}) - 9)}^{3}}$

Step 8.5.3.1.2

Factor $3$ out of $- 9$ .

$- \frac{98 t^{3}}{9 {(t^{2} - 3)}^{4}} + \frac{98 t}{{(3 t^{2} + 3 \cdot - 3)}^{3}}$

Step 8.5.3.1.3

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

$- \frac{98 t^{3}}{9 {(t^{2} - 3)}^{4}} + \frac{98 t}{{(3 (t^{2} - 3))}^{3}}$

Step 8.5.3.2

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

$- \frac{98 t^{3}}{9 {(t^{2} - 3)}^{4}} + \frac{98 t}{3^{3} {(t^{2} - 3)}^{3}}$

Step 8.5.3.3

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

$- \frac{98 t^{3}}{9 {(t^{2} - 3)}^{4}} + \frac{98 t}{27 {(t^{2} - 3)}^{3}}$