Calculus Examples

y = x^{2} + 4 x - 3

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

Step 1

Set

y

$y$ as a function of

x

$x$ .

f (x) = x^{2} + 4 x - 3

$f (x) = x^{2} + 4 x - 3$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of

x^{2} + 4 x - 3

$x^{2} + 4 x - 3$ with respect to

x

$x$ is

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$ .

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$

Step 2.1.2

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

$n = 2$ .

2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]

$2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$

2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]

$2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$

Step 2.2

Evaluate

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

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

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Step 2.2.1

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4 x

$4 x$ with respect to

x

$x$ is

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

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

2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]

$2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 2.2.2

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 = 1

$n = 1$ .

2 x + 4 \cdot 1 + \frac{d}{d x} [- 3]

$2 x + 4 \cdot 1 + \frac{d}{d x} [- 3]$

Step 2.2.3

Multiply

4

$4$ by

1

$1$ .

2 x + 4 + \frac{d}{d x} [- 3]

$2 x + 4 + \frac{d}{d x} [- 3]$

2 x + 4 + \frac{d}{d x} [- 3]

$2 x + 4 + \frac{d}{d x} [- 3]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since

- 3

$- 3$ is constant with respect to

x

$x$ , the derivative of

- 3

$- 3$ with respect to

x

$x$ is

0

$0$ .

2 x + 4 + 0

$2 x + 4 + 0$

Step 2.3.2

Add

2 x + 4

$2 x + 4$ and

0

$0$ .

2 x + 4

$2 x + 4$

2 x + 4

$2 x + 4$

2 x + 4

$2 x + 4$

Step 3

Set the derivative equal to

0

$0$ then solve the equation

2 x + 4 = 0

$2 x + 4 = 0$ .

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Step 3.1

Subtract

4

$4$ from both sides of the equation.

2 x = - 4

$2 x = - 4$

Step 3.2

Divide each term in

2 x = - 4

$2 x = - 4$ by

2

$2$ and simplify.

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Step 3.2.1

Divide each term in

2 x = - 4

$2 x = - 4$ by

2

$2$ .

\frac{2 x}{2} = \frac{- 4}{2}

$\frac{2 x}{2} = \frac{- 4}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

2

$2$ .

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Step 3.2.2.1.1

Cancel the common factor.

\frac{2 x}{2} = \frac{- 4}{2}

$\frac{2 x}{2} = \frac{- 4}{2}$

Step 3.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 4}{2}

$x = \frac{- 4}{2}$

x = \frac{- 4}{2}

$x = \frac{- 4}{2}$

x = \frac{- 4}{2}

$x = \frac{- 4}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide

- 4

$- 4$ by

2

$2$ .

x = - 2

$x = - 2$

x = - 2

$x = - 2$

x = - 2

$x = - 2$

x = - 2

$x = - 2$

Step 4

Solve the original function

f (x) = x^{2} + 4 x - 3

$f (x) = x^{2} + 4 x - 3$ at

x = - 2

$x = - 2$ .

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Step 4.1

Replace the variable

x

$x$ with

- 2

$- 2$ in the expression.

f (- 2) = {(- 2)}^{2} + 4 (- 2) - 3

$f (- 2) = {(- 2)}^{2} + 4 (- 2) - 3$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

f (- 2) = 4 + 4 (- 2) - 3

$f (- 2) = 4 + 4 (- 2) - 3$

Step 4.2.1.2

Multiply

4

$4$ by

- 2

$- 2$ .

f (- 2) = 4 - 8 - 3

$f (- 2) = 4 - 8 - 3$

f (- 2) = 4 - 8 - 3

$f (- 2) = 4 - 8 - 3$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract

8

$8$ from

4

$4$ .

f (- 2) = - 4 - 3

$f (- 2) = - 4 - 3$

Step 4.2.2.2

Subtract

3

$3$ from

- 4

$- 4$ .

f (- 2) = - 7

$f (- 2) = - 7$

f (- 2) = - 7

$f (- 2) = - 7$

Step 4.2.3

The final answer is

- 7

$- 7$ .

- 7

$- 7$

- 7

$- 7$

- 7

$- 7$

Step 5

The horizontal tangent line on function

f (x) = x^{2} + 4 x - 3

$f (x) = x^{2} + 4 x - 3$ is

y = - 7

$y = - 7$ .

y = - 7

$y = - 7$

Step 6

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