Calculus Examples

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

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = - 2 {(x + h)}^{2} - 3 (x + h) - 1$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = - 2 ((x + h) (x + h)) - 3 (x + h) - 1$

Step 2.1.2.1.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = - 2 (x (x + h) + h (x + h)) - 3 (x + h) - 1$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = - 2 (x \cdot x + x h + h (x + h)) - 3 (x + h) - 1$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = - 2 (x \cdot x + x h + h x + h \cdot h) - 3 (x + h) - 1$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = - 2 (x^{2} + x h + h x + h^{2}) - 3 (x + h) - 1$

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = - 2 (x^{2} + h x + h x + h^{2}) - 3 (x + h) - 1$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = - 2 (x^{2} + 2 h x + h^{2}) - 3 (x + h) - 1$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = - 2 x^{2} - 2 (2 h x) - 2 h^{2} - 3 (x + h) - 1$

Step 2.1.2.1.5

Multiply $2$ by $- 2$ .

$f (x + h) = - 2 x^{2} - 4 (h x) - 2 h^{2} - 3 (x + h) - 1$

Step 2.1.2.1.6

Apply the distributive property.

$f (x + h) = - 2 x^{2} - 4 h x - 2 h^{2} - 3 x - 3 h - 1$

Step 2.1.2.2

The final answer is $- 2 x^{2} - 4 h x - 2 h^{2} - 3 x - 3 h - 1$ .

$- 2 x^{2} - 4 h x - 2 h^{2} - 3 x - 3 h - 1$

Step 2.2

Reorder.

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Step 2.2.1

Move $- 3 x$ .

$- 2 x^{2} - 4 h x - 2 h^{2} - 3 h - 3 x - 1$

Step 2.2.2

Move $- 2 x^{2}$ .

$- 4 h x - 2 h^{2} - 2 x^{2} - 3 h - 3 x - 1$

Step 2.2.3

Reorder $- 4 h x$ and $- 2 h^{2}$ .

$- 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1$

Step 2.3

Find the components of the definition.

$f (x + h) = - 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1$

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

$f (x + h) = - 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1$

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

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1 - (- 2 x^{2} - 3 x - 1)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1 - (- 2 x^{2}) - (- 3 x) - - 1}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- 2$ by $- 1$ .

$\frac{- 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1 + 2 x^{2} - (- 3 x) - - 1}{h}$

Step 4.1.2.2

Multiply $- 3$ by $- 1$ .

$\frac{- 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1 + 2 x^{2} + 3 x - - 1}{h}$

Step 4.1.2.3

Multiply $- 1$ by $- 1$ .

$\frac{- 2 h^{2} - 4 h x - 2 x^{2} - 3 h - 3 x - 1 + 2 x^{2} + 3 x + 1}{h}$

Step 4.1.3

Add $- 2 x^{2}$ and $2 x^{2}$ .

$\frac{- 2 h^{2} - 4 h x - 3 h - 3 x - 1 + 0 + 3 x + 1}{h}$

Step 4.1.4

Add $- 2 h^{2}$ and $0$ .

$\frac{- 2 h^{2} - 4 h x - 3 h - 3 x - 1 + 3 x + 1}{h}$

Step 4.1.5

Add $- 3 x$ and $3 x$ .

$\frac{- 2 h^{2} - 4 h x - 3 h + 0 - 1 + 1}{h}$

Step 4.1.6

Add $- 2 h^{2}$ and $0$ .

$\frac{- 2 h^{2} - 4 h x - 3 h - 1 + 1}{h}$

Step 4.1.7

Add $- 1$ and $1$ .

$\frac{- 2 h^{2} - 4 h x - 3 h + 0}{h}$

Step 4.1.8

Add $- 2 h^{2} - 4 h x - 3 h$ and $0$ .

$\frac{- 2 h^{2} - 4 h x - 3 h}{h}$

Step 4.1.9

Factor $h$ out of $- 2 h^{2} - 4 h x - 3 h$ .

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Step 4.1.9.1

Factor $h$ out of $- 2 h^{2}$ .

$\frac{h (- 2 h) - 4 h x - 3 h}{h}$

Step 4.1.9.2

Factor $h$ out of $- 4 h x$ .

$\frac{h (- 2 h) + h (- 4 x) - 3 h}{h}$

Step 4.1.9.3

Factor $h$ out of $- 3 h$ .

$\frac{h (- 2 h) + h (- 4 x) + h \cdot - 3}{h}$

Step 4.1.9.4

Factor $h$ out of $h (- 2 h) + h (- 4 x)$ .

$\frac{h (- 2 h - 4 x) + h \cdot - 3}{h}$

Step 4.1.9.5

Factor $h$ out of $h (- 2 h - 4 x) + h \cdot - 3$ .

$\frac{h (- 2 h - 4 x - 3)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{h (- 2 h - 4 x - 3)}{h}$

Step 4.2.1.2

Divide $- 2 h - 4 x - 3$ by $1$ .

$- 2 h - 4 x - 3$

Step 4.2.2

Reorder $- 2 h$ and $- 4 x$ .

$- 4 x - 2 h - 3$

Step 5