Calculus Examples

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

Step 1

Set $y$ as a function of $x$ .

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

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 $x^{2} - 2 x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3]$ .

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $- 2$ by $1$ .

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

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$2 x - 2 + 0$

Step 2.3.2

Add $2 x - 2$ and $0$ .

$2 x - 2$

Step 3

Set the derivative equal to $0$ then solve the equation $2 x - 2 = 0$ .

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Step 3.1

Add $2$ to both sides of the equation.

$2 x = 2$

Step 3.2

Divide each term in $2 x = 2$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = 2$ by $2$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $2$ by $2$ .

$x = 1$

Step 4

Solve the original function $f (x) = x^{2} - 2 x - 3$ at $x = 1$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = {(1)}^{2} - 2 \cdot 1 - 3$

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

One to any power is one.

$f (1) = 1 - 2 \cdot 1 - 3$

Step 4.2.1.2

Multiply $- 2$ by $1$ .

$f (1) = 1 - 2 - 3$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $2$ from $1$ .

$f (1) = - 1 - 3$

Step 4.2.2.2

Subtract $3$ from $- 1$ .

$f (1) = - 4$

Step 4.2.3

The final answer is $- 4$ .

$- 4$

Step 5

The horizontal tangent line on function $f (x) = x^{2} - 2 x - 3$ is $y = - 4$ .

$y = - 4$

Step 6