Calculus Examples

$f (x) = (1 - x) {(x^{2} - 8)}^{2}$ ; $(3, - 2)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = - 2$ 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 $f (x) = 1 - x$ and $g (x) = {(x^{2} - 8)}^{2}$ .

$(1 - x) \frac{d}{d x} [{(x^{2} - 8)}^{2}] + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 8$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 8$ .

$(1 - x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} - 8]) + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.2.2

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

$(1 - x) (2 u \frac{d}{d x} [x^{2} - 8]) + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.2.3

Replace all occurrences of $u$ with $x^{2} - 8$ .

$(1 - x) (2 (x^{2} - 8) \frac{d}{d x} [x^{2} - 8]) + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.3

Differentiate.

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Step 1.3.1

Move $2$ to the left of $1 - x$ .

$2 \cdot (1 - x) (x^{2} - 8) \frac{d}{d x} [x^{2} - 8] + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.3.2

By the Sum Rule, the derivative of $x^{2} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8]$ .

$2 (1 - x) (x^{2} - 8) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8]) + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.3.3

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

$2 (1 - x) (x^{2} - 8) (2 x + \frac{d}{d x} [- 8]) + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.3.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$2 (1 - x) (x^{2} - 8) (2 x + 0) + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Add $2 x$ and $0$ .

$2 (1 - x) (x^{2} - 8) (2 x) + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.3.5.2

Multiply $2$ by $2$ .

$4 (1 - x) (x^{2} - 8) x + {(x^{2} - 8)}^{2} \frac{d}{d x} [1 - x]$

Step 1.3.6

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

$4 (1 - x) (x^{2} - 8) x + {(x^{2} - 8)}^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- x])$

Step 1.3.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 (1 - x) (x^{2} - 8) x + {(x^{2} - 8)}^{2} (0 + \frac{d}{d x} [- x])$

Step 1.3.8

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

$4 (1 - x) (x^{2} - 8) x + {(x^{2} - 8)}^{2} \frac{d}{d x} [- x]$

Step 1.3.9

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

$4 (1 - x) (x^{2} - 8) x + {(x^{2} - 8)}^{2} (- \frac{d}{d x} [x])$

Step 1.3.10

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

$4 (1 - x) (x^{2} - 8) x + {(x^{2} - 8)}^{2} (- 1 \cdot 1)$

Step 1.3.11

Simplify the expression.

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Step 1.3.11.1

Multiply $- 1$ by $1$ .

$4 (1 - x) (x^{2} - 8) x + {(x^{2} - 8)}^{2} \cdot - 1$

Step 1.3.11.2

Move $- 1$ to the left of ${(x^{2} - 8)}^{2}$ .

$4 (1 - x) (x^{2} - 8) x - 1 \cdot {(x^{2} - 8)}^{2}$

Step 1.3.11.3

Rewrite $- 1 {(x^{2} - 8)}^{2}$ as $- {(x^{2} - 8)}^{2}$ .

$4 (1 - x) (x^{2} - 8) x - {(x^{2} - 8)}^{2}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$(4 \cdot 1 + 4 (- x)) (x^{2} - 8) x - {(x^{2} - 8)}^{2}$

Step 1.4.2

Multiply $4$ by $1$ .

$(4 + 4 (- x)) (x^{2} - 8) x - {(x^{2} - 8)}^{2}$

Step 1.4.3

Multiply $- 1$ by $4$ .

$(4 - 4 x) (x^{2} - 8) x - {(x^{2} - 8)}^{2}$

Step 1.4.4

Factor $x^{2} - 8$ out of $(4 - 4 x) (x^{2} - 8) x - {(x^{2} - 8)}^{2}$ .

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Step 1.4.4.1

Factor $x^{2} - 8$ out of $(4 - 4 x) (x^{2} - 8) x$ .

$(x^{2} - 8) ((4 - 4 x) x) - {(x^{2} - 8)}^{2}$

Step 1.4.4.2

Factor $x^{2} - 8$ out of $- {(x^{2} - 8)}^{2}$ .

$(x^{2} - 8) ((4 - 4 x) x) + (x^{2} - 8) (- (x^{2} - 8))$

Step 1.4.4.3

Factor $x^{2} - 8$ out of $(x^{2} - 8) ((4 - 4 x) x) + (x^{2} - 8) (- (x^{2} - 8))$ .

$(x^{2} - 8) ((4 - 4 x) x - (x^{2} - 8))$

Step 1.4.5

Reorder the factors of $(x^{2} - 8) ((4 - 4 x) x - (x^{2} - 8))$ .

$((4 - 4 x) x - (x^{2} - 8)) (x^{2} - 8)$

Step 1.4.6

Simplify each term.

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Step 1.4.6.1

Apply the distributive property.

$(4 x - 4 x \cdot x - (x^{2} - 8)) (x^{2} - 8)$

Step 1.4.6.2

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

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Step 1.4.6.2.1

Move $x$ .

$(4 x - 4 (x \cdot x) - (x^{2} - 8)) (x^{2} - 8)$

Step 1.4.6.2.2

Multiply $x$ by $x$ .

$(4 x - 4 x^{2} - (x^{2} - 8)) (x^{2} - 8)$

Step 1.4.6.3

Apply the distributive property.

$(4 x - 4 x^{2} - x^{2} - - 8) (x^{2} - 8)$

Step 1.4.6.4

Multiply $- 1$ by $- 8$ .

$(4 x - 4 x^{2} - x^{2} + 8) (x^{2} - 8)$

Step 1.4.7

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

$(4 x - 5 x^{2} + 8) (x^{2} - 8)$

Step 1.4.8

Expand $(4 x - 5 x^{2} + 8) (x^{2} - 8)$ by multiplying each term in the first expression by each term in the second expression.

$4 x \cdot x^{2} + 4 x \cdot - 8 - 5 x^{2} x^{2} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9

Simplify each term.

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Step 1.4.9.1

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

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Step 1.4.9.1.1

Move $x^{2}$ .

$4 (x^{2} x) + 4 x \cdot - 8 - 5 x^{2} x^{2} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.1.2

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

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Step 1.4.9.1.2.1

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

$4 (x^{2} x^{1}) + 4 x \cdot - 8 - 5 x^{2} x^{2} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.1.2.2

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

$4 x^{2 + 1} + 4 x \cdot - 8 - 5 x^{2} x^{2} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.1.3

Add $2$ and $1$ .

$4 x^{3} + 4 x \cdot - 8 - 5 x^{2} x^{2} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.2

Multiply $- 8$ by $4$ .

$4 x^{3} - 32 x - 5 x^{2} x^{2} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.3

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

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Step 1.4.9.3.1

Move $x^{2}$ .

$4 x^{3} - 32 x - 5 (x^{2} x^{2}) - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.3.2

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

$4 x^{3} - 32 x - 5 x^{2 + 2} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.3.3

Add $2$ and $2$ .

$4 x^{3} - 32 x - 5 x^{4} - 5 x^{2} \cdot - 8 + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.4

Multiply $- 8$ by $- 5$ .

$4 x^{3} - 32 x - 5 x^{4} + 40 x^{2} + 8 x^{2} + 8 \cdot - 8$

Step 1.4.9.5

Multiply $8$ by $- 8$ .

$4 x^{3} - 32 x - 5 x^{4} + 40 x^{2} + 8 x^{2} - 64$

Step 1.4.10

Add $40 x^{2}$ and $8 x^{2}$ .

$4 x^{3} - 32 x - 5 x^{4} + 48 x^{2} - 64$

Step 1.5

Evaluate the derivative at $x = 3$ .

$4 {(3)}^{3} - 32 \cdot 3 - 5 {(3)}^{4} + 48 {(3)}^{2} - 64$

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

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

$4 \cdot 27 - 32 \cdot 3 - 5 {(3)}^{4} + 48 {(3)}^{2} - 64$

Step 1.6.1.2

Multiply $4$ by $27$ .

$108 - 32 \cdot 3 - 5 {(3)}^{4} + 48 {(3)}^{2} - 64$

Step 1.6.1.3

Multiply $- 32$ by $3$ .

$108 - 96 - 5 {(3)}^{4} + 48 {(3)}^{2} - 64$

Step 1.6.1.4

Raise $3$ to the power of $4$ .

$108 - 96 - 5 \cdot 81 + 48 {(3)}^{2} - 64$

Step 1.6.1.5

Multiply $- 5$ by $81$ .

$108 - 96 - 405 + 48 {(3)}^{2} - 64$

Step 1.6.1.6

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

$108 - 96 - 405 + 48 \cdot 9 - 64$

Step 1.6.1.7

Multiply $48$ by $9$ .

$108 - 96 - 405 + 432 - 64$

Step 1.6.2

Simplify by adding and subtracting.

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Step 1.6.2.1

Subtract $96$ from $108$ .

$12 - 405 + 432 - 64$

Step 1.6.2.2

Subtract $405$ from $12$ .

$- 393 + 432 - 64$

Step 1.6.2.3

Add $- 393$ and $432$ .

$39 - 64$

Step 1.6.2.4

Subtract $64$ from $39$ .

$- 25$

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 $- 25$ and a given point $(3, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = - 25 \cdot (x - (3))$

Step 2.2

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

$y + 2 = - 25 \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 25 \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y + 2 = 0 + 0 - 25 \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 2 = - 25 \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y + 2 = - 25 x - 25 \cdot - 3$

Step 2.3.1.4

Multiply $- 25$ by $- 3$ .

$y + 2 = - 25 x + 75$

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 $2$ from both sides of the equation.

$y = - 25 x + 75 - 2$

Step 2.3.2.2

Subtract $2$ from $75$ .

$y = - 25 x + 73$

Step 3