Calculus Examples

$f (x) = (x - 2) (x^{2} + 4)$ ,

$(1, - 5)$

Step 1

Find the first derivative and evaluate at

$x = 1$ and

$y = - 5$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Product Rule which states that

$\frac{d}{d x} [f (x) g (x)]$ is

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

$f (x) = x - 2$ and

$g (x) = x^{2} + 4$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of

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

$x$ is

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

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

Step 1.2.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

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

Step 1.2.3

Since

$4$ is constant with respect to

$x$ , the derivative of

$4$ with respect to

$x$ is

$0$ .

$(x - 2) (2 x + 0) + (x^{2} + 4) \frac{d}{d x} [x - 2]$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add

$2 x$ and

$0$ .

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

Step 1.2.4.2

Move

$2$ to the left of

$x - 2$ .

$2 \cdot (x - 2) x + (x^{2} + 4) \frac{d}{d x} [x - 2]$

Step 1.2.5

By the Sum Rule, the derivative of

$x - 2$ with respect to

$x$ is

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

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

Step 1.2.6

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$2 (x - 2) x + (x^{2} + 4) (1 + \frac{d}{d x} [- 2])$

Step 1.2.7

Since

$- 2$ is constant with respect to

$x$ , the derivative of

$- 2$ with respect to

$x$ is

$0$ .

$2 (x - 2) x + (x^{2} + 4) (1 + 0)$

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add

$1$ and

$0$ .

$2 (x - 2) x + (x^{2} + 4) \cdot 1$

Step 1.2.8.2

Multiply

$x^{2} + 4$ by

$1$ .

$2 (x - 2) x + x^{2} + 4$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$(2 x + 2 \cdot - 2) x + x^{2} + 4$

Step 1.3.2

Apply the distributive property.

$2 x \cdot x + 2 \cdot - 2 x + x^{2} + 4$

Step 1.3.3

Combine terms.

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Step 1.3.3.1

Raise

$x$ to the power of

$1$ .

$2 (x^{1} x) + 2 \cdot - 2 x + x^{2} + 4$

Step 1.3.3.2

Raise

$x$ to the power of

$1$ .

$2 (x^{1} x^{1}) + 2 \cdot - 2 x + x^{2} + 4$

Step 1.3.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1} + 2 \cdot - 2 x + x^{2} + 4$

Step 1.3.3.4

Add

$1$ and

$1$ .

$2 x^{2} + 2 \cdot - 2 x + x^{2} + 4$

Step 1.3.3.5

Multiply

$2$ by

$- 2$ .

$2 x^{2} - 4 x + x^{2} + 4$

Step 1.3.3.6

Add

$2 x^{2}$ and

$x^{2}$ .

$3 x^{2} - 4 x + 4$

Step 1.4

Evaluate the derivative at

$x = 1$ .

$3 {(1)}^{2} - 4 \cdot 1 + 4$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$3 \cdot 1 - 4 \cdot 1 + 4$

Step 1.5.1.2

Multiply

$3$ by

$1$ .

$3 - 4 \cdot 1 + 4$

Step 1.5.1.3

Multiply

$- 4$ by

$1$ .

$3 - 4 + 4$

Step 1.5.2

Simplify by adding and subtracting.

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Step 1.5.2.1

Subtract

$4$ from

$3$ .

$- 1 + 4$

Step 1.5.2.2

Add

$- 1$ and

$4$ .

$3$

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

$3$ and a given point

$(1, - 5)$ 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

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

$y - (- 5) = 3 \cdot (x - (1))$

Step 2.2

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

$y + 5 = 3 \cdot (x - 1)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$3 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y + 5 = 0 + 0 + 3 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 5 = 3 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y + 5 = 3 x + 3 \cdot - 1$

Step 2.3.1.4

Multiply

$3$ by

$- 1$ .

$y + 5 = 3 x - 3$

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

Subtract

$5$ from both sides of the equation.

$y = 3 x - 3 - 5$

Step 2.3.2.2

Subtract

$5$ from

$- 3$ .

$y = 3 x - 8$

Step 3