Calculus Examples

$f (x) = x^{3} {(x - 9)}^{2}$

Step 1

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

$\frac{d}{d x} [x^{3} ((x - 9) (x - 9))]$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [x^{3} (x (x - 9) - 9 (x - 9))]$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $- 9$ to the left of $x$ .

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

Step 3.1.3

Multiply $- 9$ by $- 9$ .

$\frac{d}{d x} [x^{3} (x^{2} - 9 x - 9 x + 81)]$

Step 3.2

Subtract $9 x$ from $- 9 x$ .

$\frac{d}{d x} [x^{3} (x^{2} - 18 x + 81)]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3}$ and $g (x) = x^{2} - 18 x + 81$ .

$x^{3} \frac{d}{d x} [x^{2} - 18 x + 81] + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x^{2} - 18 x + 81$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [81]$ .

$x^{3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [81]) + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5.2

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

$x^{3} (2 x + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [81]) + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5.3

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x$ with respect to $x$ is $- 18 \frac{d}{d x} [x]$ .

$x^{3} (2 x - 18 \frac{d}{d x} [x] + \frac{d}{d x} [81]) + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5.4

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

$x^{3} (2 x - 18 \cdot 1 + \frac{d}{d x} [81]) + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5.5

Multiply $- 18$ by $1$ .

$x^{3} (2 x - 18 + \frac{d}{d x} [81]) + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5.6

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$x^{3} (2 x - 18 + 0) + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5.7

Add $2 x - 18$ and $0$ .

$x^{3} (2 x - 18) + (x^{2} - 18 x + 81) \frac{d}{d x} [x^{3}]$

Step 5.8

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

$x^{3} (2 x - 18) + (x^{2} - 18 x + 81) (3 x^{2})$

Step 5.9

Move $3$ to the left of $x^{2} - 18 x + 81$ .

$x^{3} (2 x - 18) + 3 \cdot (x^{2} - 18 x + 81) x^{2}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$x^{3} (2 x) + x^{3} \cdot - 18 + 3 (x^{2} - 18 x + 81) x^{2}$

Step 6.2

Apply the distributive property.

$x^{3} (2 x) + x^{3} \cdot - 18 + (3 x^{2} + 3 (- 18 x) + 3 \cdot 81) x^{2}$

Step 6.3

Apply the distributive property.

$x^{3} (2 x) + x^{3} \cdot - 18 + 3 x^{2} x^{2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4

Combine terms.

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Step 6.4.1

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

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Step 6.4.1.1

Move $x$ .

$x \cdot x^{3} \cdot 2 + x^{3} \cdot - 18 + 3 x^{2} x^{2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.1.2

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

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Step 6.4.1.2.1

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

$x^{1} x^{3} \cdot 2 + x^{3} \cdot - 18 + 3 x^{2} x^{2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.1.2.2

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

$x^{1 + 3} \cdot 2 + x^{3} \cdot - 18 + 3 x^{2} x^{2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.1.3

Add $1$ and $3$ .

$x^{4} \cdot 2 + x^{3} \cdot - 18 + 3 x^{2} x^{2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.2

Move $2$ to the left of $x^{4}$ .

$2 \cdot x^{4} + x^{3} \cdot - 18 + 3 x^{2} x^{2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.3

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

$2 x^{4} - 18 \cdot x^{3} + 3 x^{2} x^{2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.4

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

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Step 6.4.4.1

Move $x^{2}$ .

$2 x^{4} - 18 x^{3} + 3 (x^{2} x^{2}) + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.4.2

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

$2 x^{4} - 18 x^{3} + 3 x^{2 + 2} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.4.3

Add $2$ and $2$ .

$2 x^{4} - 18 x^{3} + 3 x^{4} + 3 (- 18 x) x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.5

Multiply $- 18$ by $3$ .

$2 x^{4} - 18 x^{3} + 3 x^{4} - 54 x \cdot x^{2} + 3 \cdot 81 x^{2}$

Step 6.4.6

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

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Step 6.4.6.1

Move $x^{2}$ .

$2 x^{4} - 18 x^{3} + 3 x^{4} - 54 (x^{2} x) + 3 \cdot 81 x^{2}$

Step 6.4.6.2

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

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Step 6.4.6.2.1

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

$2 x^{4} - 18 x^{3} + 3 x^{4} - 54 (x^{2} x^{1}) + 3 \cdot 81 x^{2}$

Step 6.4.6.2.2

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

$2 x^{4} - 18 x^{3} + 3 x^{4} - 54 x^{2 + 1} + 3 \cdot 81 x^{2}$

Step 6.4.6.3

Add $2$ and $1$ .

$2 x^{4} - 18 x^{3} + 3 x^{4} - 54 x^{3} + 3 \cdot 81 x^{2}$

Step 6.4.7

Multiply $3$ by $81$ .

$2 x^{4} - 18 x^{3} + 3 x^{4} - 54 x^{3} + 243 x^{2}$

Step 6.4.8

Add $2 x^{4}$ and $3 x^{4}$ .

$5 x^{4} - 18 x^{3} - 54 x^{3} + 243 x^{2}$

Step 6.4.9

Subtract $54 x^{3}$ from $- 18 x^{3}$ .

$5 x^{4} - 72 x^{3} + 243 x^{2}$