Calculus Examples

$y = x^{2} - 6 x + 9$ , at $x = 2$

Step 1

Find the corresponding $y$ -value to $x = 2$ .

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Step 1.1

Substitute $2$ in for $x$ .

$y = {(2)}^{2} - 6 \cdot 2 + 9$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2^{2} - 6 \cdot 2 + 9$

Step 1.2.2

Remove parentheses.

$y = {(2)}^{2} - 6 \cdot 2 + 9$

Step 1.2.3

Simplify ${(2)}^{2} - 6 \cdot 2 + 9$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

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

$y = 4 - 6 \cdot 2 + 9$

Step 1.2.3.1.2

Multiply $- 6$ by $2$ .

$y = 4 - 12 + 9$

Step 1.2.3.2

Simplify by adding and subtracting.

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Step 1.2.3.2.1

Subtract $12$ from $4$ .

$y = - 8 + 9$

Step 1.2.3.2.2

Add $- 8$ and $9$ .

$y = 1$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = 1$ to find the slope of the tangent line.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{2} - 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$

Step 2.1.2

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

$2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$

Step 2.2

Evaluate $\frac{d}{d x} [- 6 x]$ .

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Step 2.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [9]$

Step 2.2.2

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

$2 x - 6 \cdot 1 + \frac{d}{d x} [9]$

Step 2.2.3

Multiply $- 6$ by $1$ .

$2 x - 6 + \frac{d}{d x} [9]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 x - 6 + 0$

Step 2.3.2

Add $2 x - 6$ and $0$ .

$2 x - 6$

Step 2.4

Evaluate the derivative at $x = 2$ .

$2 (2) - 6$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $2$ by $2$ .

$4 - 6$

Step 2.5.2

Subtract $6$ from $4$ .

$- 2$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

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

$y - (1) = - 2 \cdot (x - (2))$

Step 3.2

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

$y - 1 = - 2 \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 2 \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y - 1 = 0 + 0 - 2 \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 1 = - 2 \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - 1 = - 2 x - 2 \cdot - 2$

Step 3.3.1.4

Multiply $- 2$ by $- 2$ .

$y - 1 = - 2 x + 4$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $1$ to both sides of the equation.

$y = - 2 x + 4 + 1$

Step 3.3.2.2

Add $4$ and $1$ .

$y = - 2 x + 5$

Step 4