Calculus Examples

$f' (x) = 9 x^{2} + 4 x - 10$ , $x = - 8$

Step 1

Find the corresponding $y$ -value to $x = - 8$ .

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Step 1.1

Substitute $- 8$ in for $x$ .

$y = 9 {(- 8)}^{2} + 4 (- 8) - 10$

Step 1.2

Simplify $9 {(- 8)}^{2} + 4 (- 8) - 10$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Raise $- 8$ to the power of $2$ .

$y = 9 \cdot 64 + 4 (- 8) - 10$

Step 1.2.1.2

Multiply $9$ by $64$ .

$y = 576 + 4 (- 8) - 10$

Step 1.2.1.3

Multiply $4$ by $- 8$ .

$y = 576 - 32 - 10$

Step 1.2.2

Simplify by subtracting numbers.

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Step 1.2.2.1

Subtract $32$ from $576$ .

$y = 544 - 10$

Step 1.2.2.2

Subtract $10$ from $544$ .

$y = 534$

Step 2

Find the first derivative and evaluate at $x = - 8$ and $y = 534$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{2} + 4 x - 10$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 10]$ .

$\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 10]$

Step 2.2

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

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Step 2.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 10]$

Step 2.2.2

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

$9 (2 x) + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 10]$

Step 2.2.3

Multiply $2$ by $9$ .

$18 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 10]$

Step 2.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$18 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 10]$

Step 2.3.2

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

$18 x + 4 \cdot 1 + \frac{d}{d x} [- 10]$

Step 2.3.3

Multiply $4$ by $1$ .

$18 x + 4 + \frac{d}{d x} [- 10]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$18 x + 4 + 0$

Step 2.4.2

Add $18 x + 4$ and $0$ .

$18 x + 4$

Step 2.5

Evaluate the derivative at $x = - 8$ .

$18 (- 8) + 4$

Step 2.6

Simplify.

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Step 2.6.1

Multiply $18$ by $- 8$ .

$- 144 + 4$

Step 2.6.2

Add $- 144$ and $4$ .

$- 140$

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 $- 140$ and a given point $(- 8, 534)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (534) = - 140 \cdot (x - (- 8))$

Step 3.2

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

$y - 534 = - 140 \cdot (x + 8)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 140 \cdot (x + 8)$ .

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Step 3.3.1.1

Rewrite.

$y - 534 = 0 + 0 - 140 \cdot (x + 8)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 534 = - 140 \cdot (x + 8)$

Step 3.3.1.3

Apply the distributive property.

$y - 534 = - 140 x - 140 \cdot 8$

Step 3.3.1.4

Multiply $- 140$ by $8$ .

$y - 534 = - 140 x - 1120$

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 $534$ to both sides of the equation.

$y = - 140 x - 1120 + 534$

Step 3.3.2.2

Add $- 1120$ and $534$ .

$y = - 140 x - 586$

Step 4