Calculus Examples

y = 2 x^{3} - x^{2} + 1

$y = 2 x^{3} - x^{2} + 1$ ,

(2, 13)

$(2, 13)$

Step 1

Find the first derivative and evaluate at

x = 2

$x = 2$ and

y = 13

$y = 13$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of

2 x^{3} - x^{2} + 1

$2 x^{3} - x^{2} + 1$ with respect to

x

$x$ is

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

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

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

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

Step 1.2

Evaluate

\frac{d}{d x} [2 x^{3}]

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

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Step 1.2.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x^{3}

$2 x^{3}$ with respect to

x

$x$ is

2 \frac{d}{d x} [x^{3}]

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

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

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

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 = 3

$n = 3$ .

2 (3 x^{2}) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [1]

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

Step 1.2.3

Multiply

3

$3$ by

2

$2$ .

6 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [1]

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

6 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [1]

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

Step 1.3

Evaluate

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

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

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Step 1.3.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- x^{2}

$- x^{2}$ with respect to

x

$x$ is

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

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

6 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]

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

Step 1.3.2

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

$n = 2$ .

6 x^{2} - (2 x) + \frac{d}{d x} [1]

$6 x^{2} - (2 x) + \frac{d}{d x} [1]$

Step 1.3.3

Multiply

2

$2$ by

- 1

$- 1$ .

6 x^{2} - 2 x + \frac{d}{d x} [1]

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

6 x^{2} - 2 x + \frac{d}{d x} [1]

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

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

6 x^{2} - 2 x + 0

$6 x^{2} - 2 x + 0$

Step 1.4.2

Add

6 x^{2} - 2 x

$6 x^{2} - 2 x$ and

0

$0$ .

6 x^{2} - 2 x

$6 x^{2} - 2 x$

6 x^{2} - 2 x

$6 x^{2} - 2 x$

Step 1.5

Evaluate the derivative at

x = 2

$x = 2$ .

6 {(2)}^{2} - 2 \cdot 2

$6 {(2)}^{2} - 2 \cdot 2$

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Raise

2

$2$ to the power of

2

$2$ .

6 \cdot 4 - 2 \cdot 2

$6 \cdot 4 - 2 \cdot 2$

Step 1.6.1.2

Multiply

6

$6$ by

4

$4$ .

24 - 2 \cdot 2

$24 - 2 \cdot 2$

Step 1.6.1.3

Multiply

- 2

$- 2$ by

2

$2$ .

24 - 4

$24 - 4$

24 - 4

$24 - 4$

Step 1.6.2

Subtract

4

$4$ from

24

$24$ .

20

$20$

20

$20$

20

$20$

Step 2

Plug the slope and point values into the point-slope formula and solve for

y

$y$ .

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Step 2.1

Use the slope

20

$20$ and a given point

(2, 13)

$(2, 13)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

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

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

y - (13) = 20 \cdot (x - (2))

$y - (13) = 20 \cdot (x - (2))$

Step 2.2

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

y - 13 = 20 \cdot (x - 2)

$y - 13 = 20 \cdot (x - 2)$

Step 2.3

Solve for

y

$y$ .

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Step 2.3.1

Simplify

20 \cdot (x - 2)

$20 \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

y - 13 = 0 + 0 + 20 \cdot (x - 2)

$y - 13 = 0 + 0 + 20 \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

y - 13 = 20 \cdot (x - 2)

$y - 13 = 20 \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

y - 13 = 20 x + 20 \cdot - 2

$y - 13 = 20 x + 20 \cdot - 2$

Step 2.3.1.4

Multiply

20

$20$ by

- 2

$- 2$ .

y - 13 = 20 x - 40

$y - 13 = 20 x - 40$

y - 13 = 20 x - 40

$y - 13 = 20 x - 40$

Step 2.3.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 2.3.2.1

Add

13

$13$ to both sides of the equation.

y = 20 x - 40 + 13

$y = 20 x - 40 + 13$

Step 2.3.2.2

Add

- 40

$- 40$ and

13

$13$ .

y = 20 x - 27

$y = 20 x - 27$

y = 20 x - 27

$y = 20 x - 27$

y = 20 x - 27

$y = 20 x - 27$

y = 20 x - 27

$y = 20 x - 27$

Step 3