Calculus Examples

$h (x) = (x^{2} - 3) (x^{3} - 2 x)$ , $(- 2, - 4)$

Step 1

Find the first derivative and evaluate at $x = - 2$ 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 $f (x) = x^{2} - 3$ and $g (x) = x^{3} - 2 x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $- 2$ by $1$ .

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

Simplify the expression.

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Step 1.2.9.1

Add $2 x$ and $0$ .

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

Step 1.2.9.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Apply the distributive property.

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

Step 1.3.5

Apply the distributive property.

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

Step 1.3.6

Combine terms.

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Step 1.3.6.1

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

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Step 1.3.6.1.1

Move $x^{2}$ .

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

Step 1.3.6.1.2

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

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

Step 1.3.6.1.3

Add $2$ and $2$ .

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

Step 1.3.6.2

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

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

Step 1.3.6.3

Multiply $3$ by $- 3$ .

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

Step 1.3.6.4

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

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

Step 1.3.6.5

Multiply $- 3$ by $- 2$ .

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

Step 1.3.6.6

Subtract $2 x^{2}$ from $- 9 x^{2}$ .

$3 x^{4} - 11 x^{2} + 6 + 2 x^{3} x + 2 (- 2 x) x$

Step 1.3.6.7

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

$3 x^{4} - 11 x^{2} + 6 + 2 (x^{1} x^{3}) + 2 (- 2 x) x$

Step 1.3.6.8

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

$3 x^{4} - 11 x^{2} + 6 + 2 x^{1 + 3} + 2 (- 2 x) x$

Step 1.3.6.9

Add $1$ and $3$ .

$3 x^{4} - 11 x^{2} + 6 + 2 x^{4} + 2 (- 2 x) x$

Step 1.3.6.10

Multiply $- 2$ by $2$ .

$3 x^{4} - 11 x^{2} + 6 + 2 x^{4} - 4 x \cdot x$

Step 1.3.6.11

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

$3 x^{4} - 11 x^{2} + 6 + 2 x^{4} - 4 (x^{1} x)$

Step 1.3.6.12

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

$3 x^{4} - 11 x^{2} + 6 + 2 x^{4} - 4 (x^{1} x^{1})$

Step 1.3.6.13

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

$3 x^{4} - 11 x^{2} + 6 + 2 x^{4} - 4 x^{1 + 1}$

Step 1.3.6.14

Add $1$ and $1$ .

$3 x^{4} - 11 x^{2} + 6 + 2 x^{4} - 4 x^{2}$

Step 1.3.6.15

Add $3 x^{4}$ and $2 x^{4}$ .

$5 x^{4} - 11 x^{2} + 6 - 4 x^{2}$

Step 1.3.6.16

Subtract $4 x^{2}$ from $- 11 x^{2}$ .

$5 x^{4} - 15 x^{2} + 6$

Step 1.4

Evaluate the derivative at $x = - 2$ .

$5 {(- 2)}^{4} - 15 {(- 2)}^{2} + 6$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $- 2$ to the power of $4$ .

$5 \cdot 16 - 15 {(- 2)}^{2} + 6$

Step 1.5.1.2

Multiply $5$ by $16$ .

$80 - 15 {(- 2)}^{2} + 6$

Step 1.5.1.3

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

$80 - 15 \cdot 4 + 6$

Step 1.5.1.4

Multiply $- 15$ by $4$ .

$80 - 60 + 6$

Step 1.5.2

Simplify by adding and subtracting.

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Step 1.5.2.1

Subtract $60$ from $80$ .

$20 + 6$

Step 1.5.2.2

Add $20$ and $6$ .

$26$

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 $26$ and a given point $(- 2, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = 26 \cdot (x - (- 2))$

Step 2.2

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

$y + 4 = 26 \cdot (x + 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $26 \cdot (x + 2)$ .

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Step 2.3.1.1

Rewrite.

$y + 4 = 0 + 0 + 26 \cdot (x + 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 4 = 26 \cdot (x + 2)$

Step 2.3.1.3

Apply the distributive property.

$y + 4 = 26 x + 26 \cdot 2$

Step 2.3.1.4

Multiply $26$ by $2$ .

$y + 4 = 26 x + 52$

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

Subtract $4$ from both sides of the equation.

$y = 26 x + 52 - 4$

Step 2.3.2.2

Subtract $4$ from $52$ .

$y = 26 x + 48$

Step 3