Calculus Examples

$x^{2} {(6 - x)}^{3}$

Step 1

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 $f (x) = x^{2}$ and $g (x) = {(6 - x)}^{3}$ .

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

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 $g (x) = 6 - x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 - x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $6 - x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$x^{2} (3 {(6 - x)}^{2} (0 + \frac{d}{d x} [- x])) + {(6 - x)}^{3} \frac{d}{d x} [x^{2}]$

Step 3.3

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 3.5

Multiply $- 1$ by $3$ .

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

Step 3.6

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

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

Step 3.7

Multiply $- 3$ by $1$ .

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

Step 3.8

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

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

Step 3.9

Move $2$ to the left of ${(6 - x)}^{3}$ .

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

Step 4

Simplify.

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Step 4.1

Factor $x {(6 - x)}^{2}$ out of $x^{2} (- 3 {(6 - x)}^{2}) + 2 {(6 - x)}^{3} x$ .

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Step 4.1.1

Factor $x {(6 - x)}^{2}$ out of $x^{2} (- 3 {(6 - x)}^{2})$ .

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

Step 4.1.2

Factor $x {(6 - x)}^{2}$ out of $2 {(6 - x)}^{3} x$ .

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

Step 4.1.3

Factor $x {(6 - x)}^{2}$ out of $x {(6 - x)}^{2} (x \cdot (- 3)) + x {(6 - x)}^{2} (2 (6 - x))$ .

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

Step 4.2

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

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

Step 4.3

Rewrite ${(6 - x)}^{2}$ as $(6 - x) (6 - x)$ .

$x ((6 - x) (6 - x)) (- 3 x + 2 (6 - x))$

Step 4.4

Expand $(6 - x) (6 - x)$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

$x (6 (6 - x) - x (6 - x)) (- 3 x + 2 (6 - x))$

Step 4.4.2

Apply the distributive property.

$x (6 \cdot 6 + 6 (- x) - x (6 - x)) (- 3 x + 2 (6 - x))$

Step 4.4.3

Apply the distributive property.

$x (6 \cdot 6 + 6 (- x) - x \cdot 6 - x (- x)) (- 3 x + 2 (6 - x))$

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $6$ by $6$ .

$x (36 + 6 (- x) - x \cdot 6 - x (- x)) (- 3 x + 2 (6 - x))$

Step 4.5.1.2

Multiply $- 1$ by $6$ .

$x (36 - 6 x - x \cdot 6 - x (- x)) (- 3 x + 2 (6 - x))$

Step 4.5.1.3

Multiply $6$ by $- 1$ .

$x (36 - 6 x - 6 x - x (- x)) (- 3 x + 2 (6 - x))$

Step 4.5.1.4

Rewrite using the commutative property of multiplication.

$x (36 - 6 x - 6 x - 1 \cdot - 1 x \cdot x) (- 3 x + 2 (6 - x))$

Step 4.5.1.5

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

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Step 4.5.1.5.1

Move $x$ .

$x (36 - 6 x - 6 x - 1 \cdot - 1 (x \cdot x)) (- 3 x + 2 (6 - x))$

Step 4.5.1.5.2

Multiply $x$ by $x$ .

$x (36 - 6 x - 6 x - 1 \cdot - 1 x^{2}) (- 3 x + 2 (6 - x))$

Step 4.5.1.6

Multiply $- 1$ by $- 1$ .

$x (36 - 6 x - 6 x + 1 x^{2}) (- 3 x + 2 (6 - x))$

Step 4.5.1.7

Multiply $x^{2}$ by $1$ .

$x (36 - 6 x - 6 x + x^{2}) (- 3 x + 2 (6 - x))$

Step 4.5.2

Subtract $6 x$ from $- 6 x$ .

$x (36 - 12 x + x^{2}) (- 3 x + 2 (6 - x))$

Step 4.6

Apply the distributive property.

$(x \cdot 36 + x (- 12 x) + x \cdot x^{2}) (- 3 x + 2 (6 - x))$

Step 4.7

Simplify.

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Step 4.7.1

Move $36$ to the left of $x$ .

$(36 \cdot x + x (- 12 x) + x \cdot x^{2}) (- 3 x + 2 (6 - x))$

Step 4.7.2

Rewrite using the commutative property of multiplication.

$(36 \cdot x - 12 x \cdot x + x \cdot x^{2}) (- 3 x + 2 (6 - x))$

Step 4.7.3

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

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Step 4.7.3.1

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

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Step 4.7.3.1.1

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

$(36 \cdot x - 12 x \cdot x + x^{1} x^{2}) (- 3 x + 2 (6 - x))$

Step 4.7.3.1.2

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

$(36 \cdot x - 12 x \cdot x + x^{1 + 2}) (- 3 x + 2 (6 - x))$

Step 4.7.3.2

Add $1$ and $2$ .

$(36 \cdot x - 12 x \cdot x + x^{3}) (- 3 x + 2 (6 - x))$

Step 4.8

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

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Step 4.8.1

Move $x$ .

$(36 x - 12 (x \cdot x) + x^{3}) (- 3 x + 2 (6 - x))$

Step 4.8.2

Multiply $x$ by $x$ .

$(36 x - 12 x^{2} + x^{3}) (- 3 x + 2 (6 - x))$

Step 4.9

Simplify each term.

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Step 4.9.1

Apply the distributive property.

$(36 x - 12 x^{2} + x^{3}) (- 3 x + 2 \cdot 6 + 2 (- x))$

Step 4.9.2

Multiply $2$ by $6$ .

$(36 x - 12 x^{2} + x^{3}) (- 3 x + 12 + 2 (- x))$

Step 4.9.3

Multiply $- 1$ by $2$ .

$(36 x - 12 x^{2} + x^{3}) (- 3 x + 12 - 2 x)$

Step 4.10

Subtract $2 x$ from $- 3 x$ .

$(36 x - 12 x^{2} + x^{3}) (- 5 x + 12)$

Step 4.11

Expand $(36 x - 12 x^{2} + x^{3}) (- 5 x + 12)$ by multiplying each term in the first expression by each term in the second expression.

$36 x (- 5 x) + 36 x \cdot 12 - 12 x^{2} (- 5 x) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12

Simplify each term.

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Step 4.12.1

Rewrite using the commutative property of multiplication.

$36 \cdot - 5 x \cdot x + 36 x \cdot 12 - 12 x^{2} (- 5 x) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.2

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

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Step 4.12.2.1

Move $x$ .

$36 \cdot - 5 (x \cdot x) + 36 x \cdot 12 - 12 x^{2} (- 5 x) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.2.2

Multiply $x$ by $x$ .

$36 \cdot - 5 x^{2} + 36 x \cdot 12 - 12 x^{2} (- 5 x) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.3

Multiply $36$ by $- 5$ .

$- 180 x^{2} + 36 x \cdot 12 - 12 x^{2} (- 5 x) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.4

Multiply $12$ by $36$ .

$- 180 x^{2} + 432 x - 12 x^{2} (- 5 x) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.5

Rewrite using the commutative property of multiplication.

$- 180 x^{2} + 432 x - 12 \cdot - 5 x^{2} x - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.6

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

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Step 4.12.6.1

Move $x$ .

$- 180 x^{2} + 432 x - 12 \cdot - 5 (x \cdot x^{2}) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.6.2

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

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Step 4.12.6.2.1

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

$- 180 x^{2} + 432 x - 12 \cdot - 5 (x^{1} x^{2}) - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.6.2.2

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

$- 180 x^{2} + 432 x - 12 \cdot - 5 x^{1 + 2} - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.6.3

Add $1$ and $2$ .

$- 180 x^{2} + 432 x - 12 \cdot - 5 x^{3} - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.7

Multiply $- 12$ by $- 5$ .

$- 180 x^{2} + 432 x + 60 x^{3} - 12 x^{2} \cdot 12 + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.8

Multiply $12$ by $- 12$ .

$- 180 x^{2} + 432 x + 60 x^{3} - 144 x^{2} + x^{3} (- 5 x) + x^{3} \cdot 12$

Step 4.12.9

Rewrite using the commutative property of multiplication.

$- 180 x^{2} + 432 x + 60 x^{3} - 144 x^{2} - 5 x^{3} x + x^{3} \cdot 12$

Step 4.12.10

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

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Step 4.12.10.1

Move $x$ .

$- 180 x^{2} + 432 x + 60 x^{3} - 144 x^{2} - 5 (x \cdot x^{3}) + x^{3} \cdot 12$

Step 4.12.10.2

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

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Step 4.12.10.2.1

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

$- 180 x^{2} + 432 x + 60 x^{3} - 144 x^{2} - 5 (x^{1} x^{3}) + x^{3} \cdot 12$

Step 4.12.10.2.2

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

$- 180 x^{2} + 432 x + 60 x^{3} - 144 x^{2} - 5 x^{1 + 3} + x^{3} \cdot 12$

Step 4.12.10.3

Add $1$ and $3$ .

$- 180 x^{2} + 432 x + 60 x^{3} - 144 x^{2} - 5 x^{4} + x^{3} \cdot 12$

Step 4.12.11

Move $12$ to the left of $x^{3}$ .

$- 180 x^{2} + 432 x + 60 x^{3} - 144 x^{2} - 5 x^{4} + 12 x^{3}$

Step 4.13

Subtract $144 x^{2}$ from $- 180 x^{2}$ .

$- 324 x^{2} + 432 x + 60 x^{3} - 5 x^{4} + 12 x^{3}$

Step 4.14

Add $60 x^{3}$ and $12 x^{3}$ .

$- 324 x^{2} + 432 x - 5 x^{4} + 72 x^{3}$