Calculus Examples

$y = \frac{1}{6} \cdot {(9 x + 9)}^{3}$

Step 1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6} \cdot {(9 x + 9)}^{3}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [{(9 x + 9)}^{3}]$ .

$\frac{1}{6} \frac{d}{d x} [{(9 x + 9)}^{3}]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $9 x + 9$ .

$\frac{1}{6} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [9 x + 9])$

Step 2.2

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

$\frac{1}{6} (3 u^{2} \frac{d}{d x} [9 x + 9])$

Step 2.3

Replace all occurrences of $u$ with $9 x + 9$ .

$\frac{1}{6} (3 {(9 x + 9)}^{2} \frac{d}{d x} [9 x + 9])$

Step 3

Differentiate.

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Step 3.1

Combine $3$ and $\frac{1}{6}$ .

$\frac{3}{6} ({(9 x + 9)}^{2} \frac{d}{d x} [9 x + 9])$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine ${(9 x + 9)}^{2}$ and $\frac{3}{6}$ .

$\frac{{(9 x + 9)}^{2} \cdot 3}{6} \frac{d}{d x} [9 x + 9]$

Step 3.2.2

Move $3$ to the left of ${(9 x + 9)}^{2}$ .

$\frac{3 \cdot {(9 x + 9)}^{2}}{6} \frac{d}{d x} [9 x + 9]$

Step 3.2.3

Cancel the common factor of $3$ and $6$ .

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Step 3.2.3.1

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

$\frac{3 ({(9 x + 9)}^{2})}{6} \frac{d}{d x} [9 x + 9]$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $3$ out of $6$ .

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

Step 3.2.3.2.2

Cancel the common factor.

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

Step 3.2.3.2.3

Rewrite the expression.

$\frac{{(9 x + 9)}^{2}}{2} \frac{d}{d x} [9 x + 9]$

Step 3.3

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

$\frac{{(9 x + 9)}^{2}}{2} (\frac{d}{d x} [9 x] + \frac{d}{d x} [9])$

Step 3.4

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$\frac{{(9 x + 9)}^{2}}{2} (9 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 3.5

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

$\frac{{(9 x + 9)}^{2}}{2} (9 \cdot 1 + \frac{d}{d x} [9])$

Step 3.6

Multiply $9$ by $1$ .

$\frac{{(9 x + 9)}^{2}}{2} (9 + \frac{d}{d x} [9])$

Step 3.7

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

$\frac{{(9 x + 9)}^{2}}{2} (9 + 0)$

Step 3.8

Combine fractions.

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Step 3.8.1

Add $9$ and $0$ .

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

Step 3.8.2

Combine $\frac{{(9 x + 9)}^{2}}{2}$ and $9$ .

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

Step 3.8.3

Move $9$ to the left of ${(9 x + 9)}^{2}$ .

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${(9 x + 9)}^{2}$ as $(9 x + 9) (9 x + 9)$ .

$\frac{9 ((9 x + 9) (9 x + 9))}{2}$

Step 4.2

Expand $(9 x + 9) (9 x + 9)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$\frac{9 (9 x (9 x + 9) + 9 (9 x + 9))}{2}$

Step 4.2.2

Apply the distributive property.

$\frac{9 (9 x (9 x) + 9 x \cdot 9 + 9 (9 x + 9))}{2}$

Step 4.2.3

Apply the distributive property.

$\frac{9 (9 x (9 x) + 9 x \cdot 9 + 9 (9 x) + 9 \cdot 9)}{2}$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{9 (9 \cdot 9 x \cdot x + 9 x \cdot 9 + 9 (9 x) + 9 \cdot 9)}{2}$

Step 4.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.2.1

Move $x$ .

$\frac{9 (9 \cdot 9 (x \cdot x) + 9 x \cdot 9 + 9 (9 x) + 9 \cdot 9)}{2}$

Step 4.3.1.2.2

Multiply $x$ by $x$ .

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

Step 4.3.1.3

Multiply $9$ by $9$ .

$\frac{9 (81 x^{2} + 9 x \cdot 9 + 9 (9 x) + 9 \cdot 9)}{2}$

Step 4.3.1.4

Multiply $9$ by $9$ .

$\frac{9 (81 x^{2} + 81 x + 9 (9 x) + 9 \cdot 9)}{2}$

Step 4.3.1.5

Multiply $9$ by $9$ .

$\frac{9 (81 x^{2} + 81 x + 81 x + 9 \cdot 9)}{2}$

Step 4.3.1.6

Multiply $9$ by $9$ .

$\frac{9 (81 x^{2} + 81 x + 81 x + 81)}{2}$

Step 4.3.2

Add $81 x$ and $81 x$ .

$\frac{9 (81 x^{2} + 162 x + 81)}{2}$

Step 4.4

Apply the distributive property.

$\frac{9 (81 x^{2}) + 9 (162 x) + 9 \cdot 81}{2}$

Step 4.5

Simplify.

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Step 4.5.1

Multiply $81$ by $9$ .

$\frac{729 x^{2} + 9 (162 x) + 9 \cdot 81}{2}$

Step 4.5.2

Multiply $162$ by $9$ .

$\frac{729 x^{2} + 1458 x + 9 \cdot 81}{2}$

Step 4.5.3

Multiply $9$ by $81$ .

$\frac{729 x^{2} + 1458 x + 729}{2}$