Calculus Examples

y = x^{3} - 8 x

$y = x^{3} - 8 x$ ,

(1, - 7)

$(1, - 7)$

Step 1

Find the first derivative and evaluate at

x = 1

$x = 1$ and

y = - 7

$y = - 7$ to find the slope of the tangent line.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of

x^{3} - 8 x

$x^{3} - 8 x$ with respect to

x

$x$ is

\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x]

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x]$ .

\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x]

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x]$

Step 1.1.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

3 x^{2} + \frac{d}{d x} [- 8 x]

$3 x^{2} + \frac{d}{d x} [- 8 x]$

3 x^{2} + \frac{d}{d x} [- 8 x]

$3 x^{2} + \frac{d}{d x} [- 8 x]$

Step 1.2

Evaluate

\frac{d}{d x} [- 8 x]

$\frac{d}{d x} [- 8 x]$ .

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Step 1.2.1

Since

- 8

$- 8$ is constant with respect to

x

$x$ , the derivative of

- 8 x

$- 8 x$ with respect to

x

$x$ is

- 8 \frac{d}{d x} [x]

$- 8 \frac{d}{d x} [x]$ .

3 x^{2} - 8 \frac{d}{d x} [x]

$3 x^{2} - 8 \frac{d}{d x} [x]$

Step 1.2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

3 x^{2} - 8 \cdot 1

$3 x^{2} - 8 \cdot 1$

Step 1.2.3

Multiply

- 8

$- 8$ by

1

$1$ .

3 x^{2} - 8

$3 x^{2} - 8$

3 x^{2} - 8

$3 x^{2} - 8$

Step 1.3

Evaluate the derivative at

x = 1

$x = 1$ .

3 {(1)}^{2} - 8

$3 {(1)}^{2} - 8$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

One to any power is one.

3 \cdot 1 - 8

$3 \cdot 1 - 8$

Step 1.4.1.2

Multiply

3

$3$ by

1

$1$ .

3 - 8

$3 - 8$

3 - 8

$3 - 8$

Step 1.4.2

Subtract

8

$8$ from

3

$3$ .

- 5

$- 5$

- 5

$- 5$

- 5

$- 5$

Step 2

Plug the slope and point values into the point-slope formula and solve for

y

$y$ .

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Step 2.1

Use the slope

- 5

$- 5$ and a given point

(1, - 7)

$(1, - 7)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

y - (- 7) = - 5 \cdot (x - (1))

$y - (- 7) = - 5 \cdot (x - (1))$

Step 2.2

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

y + 7 = - 5 \cdot (x - 1)

$y + 7 = - 5 \cdot (x - 1)$

Step 2.3

Solve for

y

$y$ .

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Step 2.3.1

Simplify

- 5 \cdot (x - 1)

$- 5 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

y + 7 = 0 + 0 - 5 \cdot (x - 1)

$y + 7 = 0 + 0 - 5 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

y + 7 = - 5 \cdot (x - 1)

$y + 7 = - 5 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

y + 7 = - 5 x - 5 \cdot - 1

$y + 7 = - 5 x - 5 \cdot - 1$

Step 2.3.1.4

Multiply

- 5

$- 5$ by

- 1

$- 1$ .

y + 7 = - 5 x + 5

$y + 7 = - 5 x + 5$

y + 7 = - 5 x + 5

$y + 7 = - 5 x + 5$

Step 2.3.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 2.3.2.1

Subtract

7

$7$ from both sides of the equation.

y = - 5 x + 5 - 7

$y = - 5 x + 5 - 7$

Step 2.3.2.2

Subtract

7

$7$ from

5

$5$ .

y = - 5 x - 2

$y = - 5 x - 2$

y = - 5 x - 2

$y = - 5 x - 2$

y = - 5 x - 2

$y = - 5 x - 2$

y = - 5 x - 2

$y = - 5 x - 2$

Step 3