Calculus Examples

$x^{4} {(x - 3)}^{2}$

Step 1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$\frac{d}{d x} [x^{4} ((x - 3) (x - 3))]$

Step 2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [x^{4} (x (x - 3) - 3 (x - 3))]$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $x$ .

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

Step 3.1.2

Move $- 3$ to the left of $x$ .

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

Step 3.1.3

Multiply $- 3$ by $- 3$ .

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

Step 3.2

Subtract $3 x$ from $- 3 x$ .

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

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 $f (x) = x^{4}$ and $g (x) = x^{2} - 6 x + 9$ .

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

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x^{2} - 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$ .

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

Step 5.2

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

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

Step 5.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

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

Step 5.4

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

$x^{4} (2 x - 6 \cdot 1 + \frac{d}{d x} [9]) + (x^{2} - 6 x + 9) \frac{d}{d x} [x^{4}]$

Step 5.5

Multiply $- 6$ by $1$ .

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

Step 5.6

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

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

Step 5.7

Add $2 x - 6$ and $0$ .

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

Step 5.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$x^{4} (2 x - 6) + (x^{2} - 6 x + 9) (4 x^{3})$

Step 5.9

Move $4$ to the left of $x^{2} - 6 x + 9$ .

$x^{4} (2 x - 6) + 4 \cdot (x^{2} - 6 x + 9) x^{3}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$x^{4} (2 x) + x^{4} \cdot - 6 + 4 (x^{2} - 6 x + 9) x^{3}$

Step 6.2

Apply the distributive property.

$x^{4} (2 x) + x^{4} \cdot - 6 + (4 x^{2} + 4 (- 6 x) + 4 \cdot 9) x^{3}$

Step 6.3

Apply the distributive property.

$x^{4} (2 x) + x^{4} \cdot - 6 + 4 x^{2} x^{3} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4

Combine terms.

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Step 6.4.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 6.4.1.1

Move $x$ .

$x \cdot x^{4} \cdot 2 + x^{4} \cdot - 6 + 4 x^{2} x^{3} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.1.2

Multiply $x$ by $x^{4}$ .

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Step 6.4.1.2.1

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

$x^{1} x^{4} \cdot 2 + x^{4} \cdot - 6 + 4 x^{2} x^{3} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.1.2.2

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

$x^{1 + 4} \cdot 2 + x^{4} \cdot - 6 + 4 x^{2} x^{3} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.1.3

Add $1$ and $4$ .

$x^{5} \cdot 2 + x^{4} \cdot - 6 + 4 x^{2} x^{3} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.2

Move $2$ to the left of $x^{5}$ .

$2 \cdot x^{5} + x^{4} \cdot - 6 + 4 x^{2} x^{3} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.3

Move $- 6$ to the left of $x^{4}$ .

$2 x^{5} - 6 \cdot x^{4} + 4 x^{2} x^{3} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.4

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 6.4.4.1

Move $x^{3}$ .

$2 x^{5} - 6 x^{4} + 4 (x^{3} x^{2}) + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.4.2

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

$2 x^{5} - 6 x^{4} + 4 x^{3 + 2} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.4.3

Add $3$ and $2$ .

$2 x^{5} - 6 x^{4} + 4 x^{5} + 4 (- 6 x) x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.5

Multiply $- 6$ by $4$ .

$2 x^{5} - 6 x^{4} + 4 x^{5} - 24 x \cdot x^{3} + 4 \cdot 9 x^{3}$

Step 6.4.6

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

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Step 6.4.6.1

Move $x^{3}$ .

$2 x^{5} - 6 x^{4} + 4 x^{5} - 24 (x^{3} x) + 4 \cdot 9 x^{3}$

Step 6.4.6.2

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

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Step 6.4.6.2.1

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

$2 x^{5} - 6 x^{4} + 4 x^{5} - 24 (x^{3} x^{1}) + 4 \cdot 9 x^{3}$

Step 6.4.6.2.2

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

$2 x^{5} - 6 x^{4} + 4 x^{5} - 24 x^{3 + 1} + 4 \cdot 9 x^{3}$

Step 6.4.6.3

Add $3$ and $1$ .

$2 x^{5} - 6 x^{4} + 4 x^{5} - 24 x^{4} + 4 \cdot 9 x^{3}$

Step 6.4.7

Multiply $4$ by $9$ .

$2 x^{5} - 6 x^{4} + 4 x^{5} - 24 x^{4} + 36 x^{3}$

Step 6.4.8

Add $2 x^{5}$ and $4 x^{5}$ .

$6 x^{5} - 6 x^{4} - 24 x^{4} + 36 x^{3}$

Step 6.4.9

Subtract $24 x^{4}$ from $- 6 x^{4}$ .

$6 x^{5} - 30 x^{4} + 36 x^{3}$