Calculus Examples

$f (x) = \frac{{(x + 4)}^{12}}{{(3 x - 9)}^{11}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = {(x + 4)}^{12}$ and $g (x) = {(3 x - 9)}^{11}$ .

$\frac{{(3 x - 9)}^{11} \frac{d}{d x} [{(x + 4)}^{12}] - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{({(3 x - 9)}^{11})}^{2}}$

Step 2

Multiply the exponents in ${({(3 x - 9)}^{11})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(3 x - 9)}^{11} \frac{d}{d x} [{(x + 4)}^{12}] - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{11 \cdot 2}}$

Step 2.2

Multiply $11$ by $2$ .

$\frac{{(3 x - 9)}^{11} \frac{d}{d x} [{(x + 4)}^{12}] - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{12}$ and $g (x) = x + 4$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $x + 4$ .

$\frac{{(3 x - 9)}^{11} (\frac{d}{d u_{1}} [{u_{1}}^{12}] \frac{d}{d x} [x + 4]) - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 12$ .

$\frac{{(3 x - 9)}^{11} (12 {u_{1}}^{11} \frac{d}{d x} [x + 4]) - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 3.3

Replace all occurrences of $u_{1}$ with $x + 4$ .

$\frac{{(3 x - 9)}^{11} (12 {(x + 4)}^{11} \frac{d}{d x} [x + 4]) - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 4

Differentiate.

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Step 4.1

Move $12$ to the left of ${(3 x - 9)}^{11}$ .

$\frac{12 \cdot {(3 x - 9)}^{11} {(x + 4)}^{11} \frac{d}{d x} [x + 4] - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 4.2

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} (\frac{d}{d x} [x] + \frac{d}{d x} [4]) - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 4.3

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

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} (1 + \frac{d}{d x} [4]) - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 4.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} (1 + 0) - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Add $1$ and $0$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} \cdot 1 - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 4.5.2

Multiply $12$ by $1$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - {(x + 4)}^{12} \frac{d}{d x} [{(3 x - 9)}^{11}]}{{(3 x - 9)}^{22}}$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{11}$ and $g (x) = 3 x - 9$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $3 x - 9$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - {(x + 4)}^{12} (\frac{d}{d u_{2}} [{u_{2}}^{11}] \frac{d}{d x} [3 x - 9])}{{(3 x - 9)}^{22}}$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 11$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - {(x + 4)}^{12} (11 {u_{2}}^{10} \frac{d}{d x} [3 x - 9])}{{(3 x - 9)}^{22}}$

Step 5.3

Replace all occurrences of $u_{2}$ with $3 x - 9$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - {(x + 4)}^{12} (11 {(3 x - 9)}^{10} \frac{d}{d x} [3 x - 9])}{{(3 x - 9)}^{22}}$

Step 6

Differentiate.

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Step 6.1

Multiply $11$ by $- 1$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} ({(3 x - 9)}^{10} \frac{d}{d x} [3 x - 9])}{{(3 x - 9)}^{22}}$

Step 6.2

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

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} ({(3 x - 9)}^{10} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]))}{{(3 x - 9)}^{22}}$

Step 6.3

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

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} ({(3 x - 9)}^{10} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]))}{{(3 x - 9)}^{22}}$

Step 6.4

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

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} ({(3 x - 9)}^{10} (3 \cdot 1 + \frac{d}{d x} [- 9]))}{{(3 x - 9)}^{22}}$

Step 6.5

Multiply $3$ by $1$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} ({(3 x - 9)}^{10} (3 + \frac{d}{d x} [- 9]))}{{(3 x - 9)}^{22}}$

Step 6.6

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

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} ({(3 x - 9)}^{10} (3 + 0))}{{(3 x - 9)}^{22}}$

Step 6.7

Simplify the expression.

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Step 6.7.1

Add $3$ and $0$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} ({(3 x - 9)}^{10} \cdot 3)}{{(3 x - 9)}^{22}}$

Step 6.7.2

Move $3$ to the left of ${(3 x - 9)}^{10}$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 11 {(x + 4)}^{12} (3 \cdot {(3 x - 9)}^{10})}{{(3 x - 9)}^{22}}$

Step 6.7.3

Multiply $3$ by $- 11$ .

$\frac{12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 33 {(x + 4)}^{12} {(3 x - 9)}^{10}}{{(3 x - 9)}^{22}}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor $3 {(3 x - 9)}^{10} {(x + 4)}^{11}$ out of $12 {(3 x - 9)}^{11} {(x + 4)}^{11} - 33 {(x + 4)}^{12} {(3 x - 9)}^{10}$ .

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Step 7.1.1.1

Factor $3 {(3 x - 9)}^{10} {(x + 4)}^{11}$ out of $12 {(3 x - 9)}^{11} {(x + 4)}^{11}$ .

$\frac{3 {(3 x - 9)}^{10} {(x + 4)}^{11} (4 (3 x - 9)) - 33 {(x + 4)}^{12} {(3 x - 9)}^{10}}{{(3 x - 9)}^{22}}$

Step 7.1.1.2

Factor $3 {(3 x - 9)}^{10} {(x + 4)}^{11}$ out of $- 33 {(x + 4)}^{12} {(3 x - 9)}^{10}$ .

$\frac{3 {(3 x - 9)}^{10} {(x + 4)}^{11} (4 (3 x - 9)) + 3 {(3 x - 9)}^{10} {(x + 4)}^{11} (- 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.1.3

Factor $3 {(3 x - 9)}^{10} {(x + 4)}^{11}$ out of $3 {(3 x - 9)}^{10} {(x + 4)}^{11} (4 (3 x - 9)) + 3 {(3 x - 9)}^{10} {(x + 4)}^{11} (- 11 (x + 4))$ .

$\frac{3 {(3 x - 9)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.2

Factor $3$ out of $3 x - 9$ .

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Step 7.1.2.1

Factor $3$ out of $3 x$ .

$\frac{3 {(3 (x) - 9)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.2.2

Factor $3$ out of $- 9$ .

$\frac{3 {(3 x + 3 \cdot - 3)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.2.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$\frac{3 {(3 (x - 3))}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.3

Apply the product rule to $3 (x - 3)$ .

$\frac{3 \cdot 3^{10} {(x - 3)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.4

Multiply $3$ by $3^{10}$ by adding the exponents.

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Step 7.1.4.1

Multiply $3$ by $3^{10}$ .

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Step 7.1.4.1.1

Raise $3$ to the power of $1$ .

$\frac{3^{1} \cdot 3^{10} {(x - 3)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.4.1.2

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

$\frac{3^{1 + 10} {(x - 3)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.4.2

Add $1$ and $10$ .

$\frac{3^{11} {(x - 3)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.5

Raise $3$ to the power of $11$ .

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (4 (3 x - 9) - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.6

Simplify each term.

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Step 7.1.6.1

Apply the distributive property.

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (4 (3 x) + 4 \cdot - 9 - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.6.2

Multiply $3$ by $4$ .

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (12 x + 4 \cdot - 9 - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.6.3

Multiply $4$ by $- 9$ .

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (12 x - 36 - 11 (x + 4))}{{(3 x - 9)}^{22}}$

Step 7.1.6.4

Apply the distributive property.

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (12 x - 36 - 11 x - 11 \cdot 4)}{{(3 x - 9)}^{22}}$

Step 7.1.6.5

Multiply $- 11$ by $4$ .

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (12 x - 36 - 11 x - 44)}{{(3 x - 9)}^{22}}$

Step 7.1.7

Subtract $11 x$ from $12 x$ .

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (x - 36 - 44)}{{(3 x - 9)}^{22}}$

Step 7.1.8

Subtract $44$ from $- 36$ .

$\frac{177147 {(x - 3)}^{10} {(x + 4)}^{11} (x - 80)}{{(3 x - 9)}^{22}}$

Step 7.2

Reorder terms.

$\frac{177147 {(x + 4)}^{11} {(x - 3)}^{10} (x - 80)}{{(3 x - 9)}^{22}}$

Step 7.3

Simplify the denominator.

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Step 7.3.1

Factor $3$ out of $3 x - 9$ .

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Step 7.3.1.1

Factor $3$ out of $3 x$ .

$\frac{177147 {(x + 4)}^{11} {(x - 3)}^{10} (x - 80)}{{(3 (x) - 9)}^{22}}$

Step 7.3.1.2

Factor $3$ out of $- 9$ .

$\frac{177147 {(x + 4)}^{11} {(x - 3)}^{10} (x - 80)}{{(3 x + 3 \cdot - 3)}^{22}}$

Step 7.3.1.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$\frac{177147 {(x + 4)}^{11} {(x - 3)}^{10} (x - 80)}{{(3 (x - 3))}^{22}}$

Step 7.3.2

Apply the product rule to $3 (x - 3)$ .

$\frac{177147 {(x + 4)}^{11} {(x - 3)}^{10} (x - 80)}{3^{22} {(x - 3)}^{22}}$

Step 7.3.3

Raise $3$ to the power of $22$ .

$\frac{177147 {(x + 4)}^{11} {(x - 3)}^{10} (x - 80)}{31381059609 {(x - 3)}^{22}}$

Step 7.4

Cancel the common factor of $177147$ and $31381059609$ .

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Step 7.4.1

Factor $177147$ out of $177147 {(x + 4)}^{11} {(x - 3)}^{10} (x - 80)$ .

$\frac{177147 ({(x + 4)}^{11} {(x - 3)}^{10} (x - 80))}{31381059609 {(x - 3)}^{22}}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor $177147$ out of $31381059609 {(x - 3)}^{22}$ .

$\frac{177147 ({(x + 4)}^{11} {(x - 3)}^{10} (x - 80))}{177147 (177147 {(x - 3)}^{22})}$

Step 7.4.2.2

Cancel the common factor.

$\frac{177147 ({(x + 4)}^{11} {(x - 3)}^{10} (x - 80))}{177147 (177147 {(x - 3)}^{22})}$

Step 7.4.2.3

Rewrite the expression.

$\frac{{(x + 4)}^{11} {(x - 3)}^{10} (x - 80)}{177147 {(x - 3)}^{22}}$

Step 7.5

Cancel the common factor of ${(x - 3)}^{10}$ and ${(x - 3)}^{22}$ .

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Step 7.5.1

Factor ${(x - 3)}^{10}$ out of ${(x + 4)}^{11} {(x - 3)}^{10} (x - 80)$ .

$\frac{{(x - 3)}^{10} ({(x + 4)}^{11} (x - 80))}{177147 {(x - 3)}^{22}}$

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor ${(x - 3)}^{10}$ out of $177147 {(x - 3)}^{22}$ .

$\frac{{(x - 3)}^{10} ({(x + 4)}^{11} (x - 80))}{{(x - 3)}^{10} (177147 {(x - 3)}^{12})}$

Step 7.5.2.2

Cancel the common factor.

$\frac{{(x - 3)}^{10} ({(x + 4)}^{11} (x - 80))}{{(x - 3)}^{10} (177147 {(x - 3)}^{12})}$

Step 7.5.2.3

Rewrite the expression.

$\frac{{(x + 4)}^{11} (x - 80)}{177147 {(x - 3)}^{12}}$