Calculus Examples

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

Step 1

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 1.1

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 $f (x) = x^{3} + 3$ and $g (x) = 3 x^{2} - 4 x + 2$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $3 x^{2} - 4 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$ .

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

Step 1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $2$ by $3$ .

$(x^{3} + 3) (6 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]) + (3 x^{2} - 4 x + 2) \frac{d}{d x} [x^{3} + 3]$

Step 1.2.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$(x^{3} + 3) (6 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [2]) + (3 x^{2} - 4 x + 2) \frac{d}{d x} [x^{3} + 3]$

Step 1.2.6

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

$(x^{3} + 3) (6 x - 4 \cdot 1 + \frac{d}{d x} [2]) + (3 x^{2} - 4 x + 2) \frac{d}{d x} [x^{3} + 3]$

Step 1.2.7

Multiply $- 4$ by $1$ .

$(x^{3} + 3) (6 x - 4 + \frac{d}{d x} [2]) + (3 x^{2} - 4 x + 2) \frac{d}{d x} [x^{3} + 3]$

Step 1.2.8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$(x^{3} + 3) (6 x - 4 + 0) + (3 x^{2} - 4 x + 2) \frac{d}{d x} [x^{3} + 3]$

Step 1.2.9

Add $6 x - 4$ and $0$ .

$(x^{3} + 3) (6 x - 4) + (3 x^{2} - 4 x + 2) \frac{d}{d x} [x^{3} + 3]$

Step 1.2.10

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

$(x^{3} + 3) (6 x - 4) + (3 x^{2} - 4 x + 2) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3])$

Step 1.2.11

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

$(x^{3} + 3) (6 x - 4) + (3 x^{2} - 4 x + 2) (3 x^{2} + \frac{d}{d x} [3])$

Step 1.2.12

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

$(x^{3} + 3) (6 x - 4) + (3 x^{2} - 4 x + 2) (3 x^{2} + 0)$

Step 1.2.13

Simplify the expression.

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Step 1.2.13.1

Add $3 x^{2}$ and $0$ .

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

Step 1.2.13.2

Move $3$ to the left of $3 x^{2} - 4 x + 2$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Apply the distributive property.

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

Step 1.3.5

Apply the distributive property.

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

Step 1.3.6

Combine terms.

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Step 1.3.6.1

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

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Step 1.3.6.1.1

Move $x$ .

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

Step 1.3.6.1.2

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

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Step 1.3.6.1.2.1

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

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

Step 1.3.6.1.2.2

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

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

Step 1.3.6.1.3

Add $1$ and $3$ .

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

Step 1.3.6.2

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

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

Step 1.3.6.3

Multiply $6$ by $3$ .

$6 x^{4} + 18 x + x^{3} \cdot - 4 + 3 \cdot - 4 + 3 (3 x^{2}) x^{2} + 3 (- 4 x) x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.4

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

$6 x^{4} + 18 x - 4 \cdot x^{3} + 3 \cdot - 4 + 3 (3 x^{2}) x^{2} + 3 (- 4 x) x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.5

Multiply $3$ by $- 4$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 3 (3 x^{2}) x^{2} + 3 (- 4 x) x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.6

Multiply $3$ by $3$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{2} x^{2} + 3 (- 4 x) x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.7

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

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Step 1.3.6.7.1

Move $x^{2}$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 (x^{2} x^{2}) + 3 (- 4 x) x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.7.2

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

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{2 + 2} + 3 (- 4 x) x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.7.3

Add $2$ and $2$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{4} + 3 (- 4 x) x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.8

Multiply $- 4$ by $3$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{4} - 12 x \cdot x^{2} + 3 \cdot 2 x^{2}$

Step 1.3.6.9

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

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Step 1.3.6.9.1

Move $x^{2}$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{4} - 12 (x^{2} x) + 3 \cdot 2 x^{2}$

Step 1.3.6.9.2

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

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Step 1.3.6.9.2.1

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

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{4} - 12 (x^{2} x^{1}) + 3 \cdot 2 x^{2}$

Step 1.3.6.9.2.2

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

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{4} - 12 x^{2 + 1} + 3 \cdot 2 x^{2}$

Step 1.3.6.9.3

Add $2$ and $1$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{4} - 12 x^{3} + 3 \cdot 2 x^{2}$

Step 1.3.6.10

Multiply $3$ by $2$ .

$6 x^{4} + 18 x - 4 x^{3} - 12 + 9 x^{4} - 12 x^{3} + 6 x^{2}$

Step 1.3.6.11

Add $6 x^{4}$ and $9 x^{4}$ .

$15 x^{4} + 18 x - 4 x^{3} - 12 - 12 x^{3} + 6 x^{2}$

Step 1.3.6.12

Subtract $12 x^{3}$ from $- 4 x^{3}$ .

$15 x^{4} + 18 x - 16 x^{3} - 12 + 6 x^{2}$

Step 1.3.7

Reorder terms.

$15 x^{4} - 16 x^{3} + 6 x^{2} + 18 x - 12$

Step 1.4

Evaluate the derivative at $x = 1$ .

$15 {(1)}^{4} - 16 {(1)}^{3} + 6 {(1)}^{2} + 18 (1) - 12$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$15 \cdot 1 - 16 {(1)}^{3} + 6 {(1)}^{2} + 18 (1) - 12$

Step 1.5.1.2

Multiply $15$ by $1$ .

$15 - 16 {(1)}^{3} + 6 {(1)}^{2} + 18 (1) - 12$

Step 1.5.1.3

One to any power is one.

$15 - 16 \cdot 1 + 6 {(1)}^{2} + 18 (1) - 12$

Step 1.5.1.4

Multiply $- 16$ by $1$ .

$15 - 16 + 6 {(1)}^{2} + 18 (1) - 12$

Step 1.5.1.5

One to any power is one.

$15 - 16 + 6 \cdot 1 + 18 (1) - 12$

Step 1.5.1.6

Multiply $6$ by $1$ .

$15 - 16 + 6 + 18 (1) - 12$

Step 1.5.1.7

Multiply $18$ by $1$ .

$15 - 16 + 6 + 18 - 12$

Step 1.5.2

Simplify by adding and subtracting.

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Step 1.5.2.1

Subtract $16$ from $15$ .

$- 1 + 6 + 18 - 12$

Step 1.5.2.2

Add $- 1$ and $6$ .

$5 + 18 - 12$

Step 1.5.2.3

Add $5$ and $18$ .

$23 - 12$

Step 1.5.2.4

Subtract $12$ from $23$ .

$11$

Step 2

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

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Step 2.1

Use the slope $11$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = 11 \cdot (x - (1))$

Step 2.2

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

$y - 4 = 11 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $11 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 + 11 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = 11 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = 11 x + 11 \cdot - 1$

Step 2.3.1.4

Multiply $11$ by $- 1$ .

$y - 4 = 11 x - 11$

Step 2.3.2

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

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Step 2.3.2.1

Add $4$ to both sides of the equation.

$y = 11 x - 11 + 4$

Step 2.3.2.2

Add $- 11$ and $4$ .

$y = 11 x - 7$

Step 3