Calculus Examples

$f (x) = x^{4} {(3 - x)}^{2}$ ; $x = 1$

Step 1

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

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Step 1.1

Substitute $1$ in for $x$ .

$y = {(1)}^{4} {(3 - (1))}^{2}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 1^{4} {(3 - (1))}^{2}$

Step 1.2.2

Remove parentheses.

$y = {(1)}^{4} {(3 - (1))}^{2}$

Step 1.2.3

Simplify ${(1)}^{4} {(3 - (1))}^{2}$ .

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Step 1.2.3.1

One to any power is one.

$y = 1 {(3 - (1))}^{2}$

Step 1.2.3.2

Multiply ${(3 - (1))}^{2}$ by $1$ .

$y = {(3 - (1))}^{2}$

Step 1.2.3.3

Multiply $- 1$ by $1$ .

$y = {(3 - 1)}^{2}$

Step 1.2.3.4

Subtract $1$ from $3$ .

$y = 2^{2}$

Step 1.2.3.5

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

$y = 4$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 4$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $3$ by $3$ .

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

Step 2.3.1.2

Multiply $- 1$ by $3$ .

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

Step 2.3.1.3

Multiply $3$ by $- 1$ .

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

Step 2.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [x^{4} (9 - 3 x - 3 x - 1 \cdot - 1 x \cdot x)]$

Step 2.3.1.5

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

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Step 2.3.1.5.1

Move $x$ .

$\frac{d}{d x} [x^{4} (9 - 3 x - 3 x - 1 \cdot - 1 (x \cdot x))]$

Step 2.3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{4} (9 - 3 x - 3 x - 1 \cdot - 1 x^{2})]$

Step 2.3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [x^{4} (9 - 3 x - 3 x + 1 x^{2})]$

Step 2.3.1.7

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

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

Step 2.3.2

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

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

Step 2.4

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

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

Step 2.5

Differentiate.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

Add $0$ and $\frac{d}{d x} [- 6 x]$ .

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

Step 2.5.4

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

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

Step 2.5.5

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

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

Step 2.5.6

Multiply $- 6$ by $1$ .

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

Step 2.5.7

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

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

Step 2.5.8

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

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

Step 2.5.9

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

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.6.4

Combine terms.

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Step 2.6.4.1

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

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

Step 2.6.4.2

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

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Step 2.6.4.2.1

Move $x$ .

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

Step 2.6.4.2.2

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

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Step 2.6.4.2.2.1

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

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

Step 2.6.4.2.2.2

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

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

Step 2.6.4.2.3

Add $1$ and $4$ .

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

Step 2.6.4.3

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

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

Step 2.6.4.4

Multiply $4$ by $9$ .

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

Step 2.6.4.5

Multiply $- 6$ by $4$ .

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

Step 2.6.4.6

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

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Step 2.6.4.6.1

Move $x^{3}$ .

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

Step 2.6.4.6.2

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

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Step 2.6.4.6.2.1

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

$- 6 x^{4} + 2 x^{5} + 36 x^{3} - 24 (x^{3} x^{1}) + 4 x^{2} x^{3}$

Step 2.6.4.6.2.2

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

$- 6 x^{4} + 2 x^{5} + 36 x^{3} - 24 x^{3 + 1} + 4 x^{2} x^{3}$

Step 2.6.4.6.3

Add $3$ and $1$ .

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

Step 2.6.4.7

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

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Step 2.6.4.7.1

Move $x^{3}$ .

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

Step 2.6.4.7.2

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

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

Step 2.6.4.7.3

Add $3$ and $2$ .

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

Step 2.6.4.8

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

$- 30 x^{4} + 2 x^{5} + 36 x^{3} + 4 x^{5}$

Step 2.6.4.9

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

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

Step 2.6.5

Reorder terms.

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

Step 2.7

Evaluate the derivative at $x = 1$ .

$6 {(1)}^{5} - 30 {(1)}^{4} + 36 {(1)}^{3}$

Step 2.8

Simplify.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

One to any power is one.

$6 \cdot 1 - 30 {(1)}^{4} + 36 {(1)}^{3}$

Step 2.8.1.2

Multiply $6$ by $1$ .

$6 - 30 {(1)}^{4} + 36 {(1)}^{3}$

Step 2.8.1.3

One to any power is one.

$6 - 30 \cdot 1 + 36 {(1)}^{3}$

Step 2.8.1.4

Multiply $- 30$ by $1$ .

$6 - 30 + 36 {(1)}^{3}$

Step 2.8.1.5

One to any power is one.

$6 - 30 + 36 \cdot 1$

Step 2.8.1.6

Multiply $36$ by $1$ .

$6 - 30 + 36$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $30$ from $6$ .

$- 24 + 36$

Step 2.8.2.2

Add $- 24$ and $36$ .

$12$

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 $12$ and a given point $(1, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = 12 \cdot (x - (1))$

Step 3.2

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

$y - 4 = 12 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $12 \cdot (x - 1)$ .

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Step 3.3.1.1

Rewrite.

$y - 4 = 0 + 0 + 12 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 4 = 12 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y - 4 = 12 x + 12 \cdot - 1$

Step 3.3.1.4

Multiply $12$ by $- 1$ .

$y - 4 = 12 x - 12$

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 $4$ to both sides of the equation.

$y = 12 x - 12 + 4$

Step 3.3.2.2

Add $- 12$ and $4$ .

$y = 12 x - 8$

Step 4