Calculus Examples

$f (x) = x^{4} {(3 - x)}^{3}$ ; $x = 1$

Step 1

Find the corresponding $y$ -value to $x = 1$ .

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Step 1.1

Substitute $1$ in for $x$ .

$y = {(1)}^{4} {(3 - (1))}^{3}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 1^{4} {(3 - (1))}^{3}$

Step 1.2.2

Remove parentheses.

$y = {(1)}^{4} {(3 - (1))}^{3}$

Step 1.2.3

Simplify ${(1)}^{4} {(3 - (1))}^{3}$ .

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Step 1.2.3.1

One to any power is one.

$y = 1 {(3 - (1))}^{3}$

Step 1.2.3.2

Multiply ${(3 - (1))}^{3}$ by $1$ .

$y = {(3 - (1))}^{3}$

Step 1.2.3.3

Multiply $- 1$ by $1$ .

$y = {(3 - 1)}^{3}$

Step 1.2.3.4

Subtract $1$ from $3$ .

$y = 2^{3}$

Step 1.2.3.5

Raise $2$ to the power of $3$ .

$y = 8$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 8$ to find the slope of the tangent line.

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Step 2.1

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 $f (x) = x^{4}$ and $g (x) = {(3 - x)}^{3}$ .

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

Step 2.2

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 $g (x) = 3 - x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $3 - x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $3 - x$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$x^{4} (3 {(3 - x)}^{2} (0 + \frac{d}{d x} [- x])) + {(3 - x)}^{3} \frac{d}{d x} [x^{4}]$

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $- 1$ by $3$ .

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

Step 2.3.6

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

$x^{4} (- 3 {(3 - x)}^{2} \cdot 1) + {(3 - x)}^{3} \frac{d}{d x} [x^{4}]$

Step 2.3.7

Multiply $- 3$ by $1$ .

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

Step 2.3.8

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

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

Step 2.3.9

Move $4$ to the left of ${(3 - x)}^{3}$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Factor $x^{3} {(3 - x)}^{2}$ out of $x^{4} (- 3 {(3 - x)}^{2}) + 4 {(3 - x)}^{3} x^{3}$ .

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Step 2.4.1.1

Factor $x^{3} {(3 - x)}^{2}$ out of $x^{4} (- 3 {(3 - x)}^{2})$ .

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

Step 2.4.1.2

Factor $x^{3} {(3 - x)}^{2}$ out of $4 {(3 - x)}^{3} x^{3}$ .

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

Step 2.4.1.3

Factor $x^{3} {(3 - x)}^{2}$ out of $x^{3} {(3 - x)}^{2} (x \cdot (- 3)) + x^{3} {(3 - x)}^{2} (4 (3 - x))$ .

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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Step 2.4.4.1

Apply the distributive property.

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

Step 2.4.4.2

Apply the distributive property.

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

Step 2.4.4.3

Apply the distributive property.

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

Step 2.4.5

Simplify and combine like terms.

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Multiply $3$ by $3$ .

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

Step 2.4.5.1.2

Multiply $- 1$ by $3$ .

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

Step 2.4.5.1.3

Multiply $3$ by $- 1$ .

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

Step 2.4.5.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.4.5.1.5

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

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Step 2.4.5.1.5.1

Move $x$ .

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

Step 2.4.5.1.5.2

Multiply $x$ by $x$ .

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

Step 2.4.5.1.6

Multiply $- 1$ by $- 1$ .

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

Step 2.4.5.1.7

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

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

Step 2.4.5.2

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

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

Step 2.4.6

Apply the distributive property.

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

Step 2.4.7

Simplify.

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Step 2.4.7.1

Move $9$ to the left of $x^{3}$ .

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

Step 2.4.7.2

Rewrite using the commutative property of multiplication.

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

Step 2.4.7.3

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

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Step 2.4.7.3.1

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

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

Step 2.4.7.3.2

Add $3$ and $2$ .

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

Step 2.4.8

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

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Step 2.4.8.1

Move $x$ .

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

Step 2.4.8.2

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

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Step 2.4.8.2.1

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

$(9 x^{3} - 6 (x^{1} x^{3}) + x^{5}) (- 3 x + 4 (3 - x))$

Step 2.4.8.2.2

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

$(9 x^{3} - 6 x^{1 + 3} + x^{5}) (- 3 x + 4 (3 - x))$

Step 2.4.8.3

Add $1$ and $3$ .

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

Step 2.4.9

Simplify each term.

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Step 2.4.9.1

Apply the distributive property.

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

Step 2.4.9.2

Multiply $4$ by $3$ .

$(9 x^{3} - 6 x^{4} + x^{5}) (- 3 x + 12 + 4 (- x))$

Step 2.4.9.3

Multiply $- 1$ by $4$ .

$(9 x^{3} - 6 x^{4} + x^{5}) (- 3 x + 12 - 4 x)$

Step 2.4.10

Subtract $4 x$ from $- 3 x$ .

$(9 x^{3} - 6 x^{4} + x^{5}) (- 7 x + 12)$

Step 2.4.11

Expand $(9 x^{3} - 6 x^{4} + x^{5}) (- 7 x + 12)$ by multiplying each term in the first expression by each term in the second expression.

$9 x^{3} (- 7 x) + 9 x^{3} \cdot 12 - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12

Simplify each term.

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Step 2.4.12.1

Rewrite using the commutative property of multiplication.

$9 \cdot - 7 x^{3} x + 9 x^{3} \cdot 12 - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.2

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

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Step 2.4.12.2.1

Move $x$ .

$9 \cdot - 7 (x \cdot x^{3}) + 9 x^{3} \cdot 12 - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.2.2

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

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Step 2.4.12.2.2.1

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

$9 \cdot - 7 (x^{1} x^{3}) + 9 x^{3} \cdot 12 - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.2.2.2

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

$9 \cdot - 7 x^{1 + 3} + 9 x^{3} \cdot 12 - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.2.3

Add $1$ and $3$ .

$9 \cdot - 7 x^{4} + 9 x^{3} \cdot 12 - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.3

Multiply $9$ by $- 7$ .

$- 63 x^{4} + 9 x^{3} \cdot 12 - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.4

Multiply $12$ by $9$ .

$- 63 x^{4} + 108 x^{3} - 6 x^{4} (- 7 x) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.5

Rewrite using the commutative property of multiplication.

$- 63 x^{4} + 108 x^{3} - 6 \cdot - 7 x^{4} x - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.6

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

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Step 2.4.12.6.1

Move $x$ .

$- 63 x^{4} + 108 x^{3} - 6 \cdot - 7 (x \cdot x^{4}) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.6.2

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

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Step 2.4.12.6.2.1

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

$- 63 x^{4} + 108 x^{3} - 6 \cdot - 7 (x^{1} x^{4}) - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.6.2.2

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

$- 63 x^{4} + 108 x^{3} - 6 \cdot - 7 x^{1 + 4} - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.6.3

Add $1$ and $4$ .

$- 63 x^{4} + 108 x^{3} - 6 \cdot - 7 x^{5} - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.7

Multiply $- 6$ by $- 7$ .

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 6 x^{4} \cdot 12 + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.8

Multiply $12$ by $- 6$ .

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 72 x^{4} + x^{5} (- 7 x) + x^{5} \cdot 12$

Step 2.4.12.9

Rewrite using the commutative property of multiplication.

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 72 x^{4} - 7 x^{5} x + x^{5} \cdot 12$

Step 2.4.12.10

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 2.4.12.10.1

Move $x$ .

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 72 x^{4} - 7 (x \cdot x^{5}) + x^{5} \cdot 12$

Step 2.4.12.10.2

Multiply $x$ by $x^{5}$ .

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Step 2.4.12.10.2.1

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

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 72 x^{4} - 7 (x^{1} x^{5}) + x^{5} \cdot 12$

Step 2.4.12.10.2.2

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

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 72 x^{4} - 7 x^{1 + 5} + x^{5} \cdot 12$

Step 2.4.12.10.3

Add $1$ and $5$ .

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 72 x^{4} - 7 x^{6} + x^{5} \cdot 12$

Step 2.4.12.11

Move $12$ to the left of $x^{5}$ .

$- 63 x^{4} + 108 x^{3} + 42 x^{5} - 72 x^{4} - 7 x^{6} + 12 x^{5}$

Step 2.4.13

Subtract $72 x^{4}$ from $- 63 x^{4}$ .

$- 135 x^{4} + 108 x^{3} + 42 x^{5} - 7 x^{6} + 12 x^{5}$

Step 2.4.14

Add $42 x^{5}$ and $12 x^{5}$ .

$- 135 x^{4} + 108 x^{3} - 7 x^{6} + 54 x^{5}$

Step 2.5

Evaluate the derivative at $x = 1$ .

$- 135 {(1)}^{4} + 108 {(1)}^{3} - 7 {(1)}^{6} + 54 {(1)}^{5}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

One to any power is one.

$- 135 \cdot 1 + 108 {(1)}^{3} - 7 {(1)}^{6} + 54 {(1)}^{5}$

Step 2.6.1.2

Multiply $- 135$ by $1$ .

$- 135 + 108 {(1)}^{3} - 7 {(1)}^{6} + 54 {(1)}^{5}$

Step 2.6.1.3

One to any power is one.

$- 135 + 108 \cdot 1 - 7 {(1)}^{6} + 54 {(1)}^{5}$

Step 2.6.1.4

Multiply $108$ by $1$ .

$- 135 + 108 - 7 {(1)}^{6} + 54 {(1)}^{5}$

Step 2.6.1.5

One to any power is one.

$- 135 + 108 - 7 \cdot 1 + 54 {(1)}^{5}$

Step 2.6.1.6

Multiply $- 7$ by $1$ .

$- 135 + 108 - 7 + 54 {(1)}^{5}$

Step 2.6.1.7

One to any power is one.

$- 135 + 108 - 7 + 54 \cdot 1$

Step 2.6.1.8

Multiply $54$ by $1$ .

$- 135 + 108 - 7 + 54$

Step 2.6.2

Simplify by adding and subtracting.

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Step 2.6.2.1

Add $- 135$ and $108$ .

$- 27 - 7 + 54$

Step 2.6.2.2

Subtract $7$ from $- 27$ .

$- 34 + 54$

Step 2.6.2.3

Add $- 34$ and $54$ .

$20$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $20$ and a given point $(1, 8)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = 20 \cdot (x - (1))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 8 = 20 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $20 \cdot (x - 1)$ .

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Step 3.3.1.1

Rewrite.

$y - 8 = 0 + 0 + 20 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 8 = 20 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y - 8 = 20 x + 20 \cdot - 1$

Step 3.3.1.4

Multiply $20$ by $- 1$ .

$y - 8 = 20 x - 20$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $8$ to both sides of the equation.

$y = 20 x - 20 + 8$

Step 3.3.2.2

Add $- 20$ and $8$ .

$y = 20 x - 12$

Step 4