Calculus Examples

$f (x) = 5 x^{2} + 6 x - 11$ , $x = - 2$

Step 1

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

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Step 1.1

Substitute $- 2$ in for $x$ .

$y = 5 {(- 2)}^{2} + 6 (- 2) - 11$

Step 1.2

Simplify $5 {(- 2)}^{2} + 6 (- 2) - 11$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

$y = 5 \cdot 4 + 6 (- 2) - 11$

Step 1.2.1.2

Multiply $5$ by $4$ .

$y = 20 + 6 (- 2) - 11$

Step 1.2.1.3

Multiply $6$ by $- 2$ .

$y = 20 - 12 - 11$

Step 1.2.2

Simplify by subtracting numbers.

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Step 1.2.2.1

Subtract $12$ from $20$ .

$y = 8 - 11$

Step 1.2.2.2

Subtract $11$ from $8$ .

$y = - 3$

Step 2

Find the first derivative and evaluate at $x = - 2$ and $y = - 3$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{2} + 6 x - 11$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 11]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 11]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 11]$

Step 2.2.2

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

$5 (2 x) + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 11]$

Step 2.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 11]$

Step 2.3

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

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Step 2.3.1

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

$10 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 11]$

Step 2.3.2

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

$10 x + 6 \cdot 1 + \frac{d}{d x} [- 11]$

Step 2.3.3

Multiply $6$ by $1$ .

$10 x + 6 + \frac{d}{d x} [- 11]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11$ with respect to $x$ is $0$ .

$10 x + 6 + 0$

Step 2.4.2

Add $10 x + 6$ and $0$ .

$10 x + 6$

Step 2.5

Evaluate the derivative at $x = - 2$ .

$10 (- 2) + 6$

Step 2.6

Simplify.

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Step 2.6.1

Multiply $10$ by $- 2$ .

$- 20 + 6$

Step 2.6.2

Add $- 20$ and $6$ .

$- 14$

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 $- 14$ and a given point $(- 2, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = - 14 \cdot (x - (- 2))$

Step 3.2

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

$y + 3 = - 14 \cdot (x + 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 14 \cdot (x + 2)$ .

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Step 3.3.1.1

Rewrite.

$y + 3 = 0 + 0 - 14 \cdot (x + 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y + 3 = - 14 \cdot (x + 2)$

Step 3.3.1.3

Apply the distributive property.

$y + 3 = - 14 x - 14 \cdot 2$

Step 3.3.1.4

Multiply $- 14$ by $2$ .

$y + 3 = - 14 x - 28$

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

Subtract $3$ from both sides of the equation.

$y = - 14 x - 28 - 3$

Step 3.3.2.2

Subtract $3$ from $- 28$ .

$y = - 14 x - 31$

Step 4