Calculus Examples

$f (x) = (x - 4) (x^{2} + 8)$ , $(1, - 27)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = - 27$ 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 $f (x) = x - 4$ and $g (x) = x^{2} + 8$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.2.4.2

Move $2$ to the left of $x - 4$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 (x - 4) x + (x^{2} + 8) (1 + 0)$

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $1$ and $0$ .

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

Step 1.2.8.2

Multiply $x^{2} + 8$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Combine terms.

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

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

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

Step 1.3.3.4

Add $1$ and $1$ .

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

Step 1.3.3.5

Multiply $2$ by $- 4$ .

$2 x^{2} - 8 x + x^{2} + 8$

Step 1.3.3.6

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

$3 x^{2} - 8 x + 8$

Step 1.4

Evaluate the derivative at $x = 1$ .

$3 {(1)}^{2} - 8 \cdot 1 + 8$

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.

$3 \cdot 1 - 8 \cdot 1 + 8$

Step 1.5.1.2

Multiply $3$ by $1$ .

$3 - 8 \cdot 1 + 8$

Step 1.5.1.3

Multiply $- 8$ by $1$ .

$3 - 8 + 8$

Step 1.5.2

Simplify by adding and subtracting.

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Step 1.5.2.1

Subtract $8$ from $3$ .

$- 5 + 8$

Step 1.5.2.2

Add $- 5$ and $8$ .

$3$

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 $3$ and a given point $(1, - 27)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 27) = 3 \cdot (x - (1))$

Step 2.2

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

$y + 27 = 3 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $3 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y + 27 = 0 + 0 + 3 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 27 = 3 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y + 27 = 3 x + 3 \cdot - 1$

Step 2.3.1.4

Multiply $3$ by $- 1$ .

$y + 27 = 3 x - 3$

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 $27$ from both sides of the equation.

$y = 3 x - 3 - 27$

Step 2.3.2.2

Subtract $27$ from $- 3$ .

$y = 3 x - 30$

Step 3