Calculus Examples

y = 4 x^{2} + 6 x

$y = 4 x^{2} + 6 x$ ,

(- 3, 18)

$(- 3, 18)$

Step 1

Find the first derivative and evaluate at

x = - 3

$x = - 3$ and

y = 18

$y = 18$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of

4 x^{2} + 6 x

$4 x^{2} + 6 x$ with respect to

x

$x$ is

\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [6 x]

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [6 x]$ .

\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [6 x]

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.2

Evaluate

\frac{d}{d x} [4 x^{2}]

$\frac{d}{d x} [4 x^{2}]$ .

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Step 1.2.1

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4 x^{2}

$4 x^{2}$ with respect to

x

$x$ is

4 \frac{d}{d x} [x^{2}]

$4 \frac{d}{d x} [x^{2}]$ .

4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x]

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x]$

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 = 2

$n = 2$ .

4 (2 x) + \frac{d}{d x} [6 x]

$4 (2 x) + \frac{d}{d x} [6 x]$

Step 1.2.3

Multiply

2

$2$ by

4

$4$ .

8 x + \frac{d}{d x} [6 x]

$8 x + \frac{d}{d x} [6 x]$

Step 1.3

Evaluate

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

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Step 1.3.1

Since

$6$ is constant with respect to

$x$ , the derivative of

$6 x$ with respect to

$x$ is

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

$8 x + 6 \frac{d}{d x} [x]$

Step 1.3.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$8 x + 6 \cdot 1$

Step 1.3.3

Multiply

$6$ by

$1$ .

$8 x + 6$

Step 1.4

Evaluate the derivative at

$x = - 3$ .

$8 (- 3) + 6$

Step 1.5

Simplify.

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Step 1.5.1

Multiply

$8$ by

$- 3$ .

$- 24 + 6$

Step 1.5.2

Add

$- 24$ and

$6$ .

$- 18$

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

$- 18$ and a given point

$(- 3, 18)$ 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

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

$y - (18) = - 18 \cdot (x - (- 3))$

Step 2.2

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

$y - 18 = - 18 \cdot (x + 3)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$- 18 \cdot (x + 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 18 = 0 + 0 - 18 \cdot (x + 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 18 = - 18 \cdot (x + 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 18 = - 18 x - 18 \cdot 3$

Step 2.3.1.4

Multiply

$- 18$ by

$3$ .

$y - 18 = - 18 x - 54$

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

$18$ to both sides of the equation.

$y = - 18 x - 54 + 18$

Step 2.3.2.2

Add

$- 54$ and

$18$ .

$y = - 18 x - 36$

Step 3

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