Calculus Examples

$y = x^{3} - 3 x + 2$ ,

$(2, 4)$

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.

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Step 1.1.1

By the Sum Rule, the derivative of

$x^{3} - 3 x + 2$ with respect to

$x$ is

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

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

Step 1.1.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 3$ .

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

Step 1.2

Evaluate

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

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Step 1.2.1

Since

$- 3$ is constant with respect to

$x$ , the derivative of

$- 3 x$ with respect to

$x$ is

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

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

Step 1.2.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.2.3

Multiply

$- 3$ by

$1$ .

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

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

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

Step 1.3.2

Add

$3 x^{2} - 3$ and

$0$ .

$3 x^{2} - 3$

Step 1.4

Evaluate the derivative at

$x = 2$ .

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

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

$2$ .

$3 \cdot 4 - 3$

Step 1.5.1.2

Multiply

$3$ by

$4$ .

$12 - 3$

Step 1.5.2

Subtract

$3$ from

$12$ .

$9$

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

$9$ 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

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

$y - (4) = 9 \cdot (x - (2))$

Step 2.2

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

$y - 4 = 9 \cdot (x - 2)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$9 \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 + 9 \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = 9 \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = 9 x + 9 \cdot - 2$

Step 2.3.1.4

Multiply

$9$ by

$- 2$ .

$y - 4 = 9 x - 18$

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.

$y = 9 x - 18 + 4$

Step 2.3.2.2

Add

$- 18$ and

$4$ .

$y = 9 x - 14$

Step 3