Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

$f (x + h) = - ((x + h) (x + h))$

Step 2.1.2.2

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

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Step 2.1.2.2.1

Apply the distributive property.

$f (x + h) = - (x (x + h) + h (x + h))$

Step 2.1.2.2.2

Apply the distributive property.

$f (x + h) = - (x \cdot x + x h + h (x + h))$

Step 2.1.2.2.3

Apply the distributive property.

$f (x + h) = - (x \cdot x + x h + h x + h \cdot h)$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.4

Apply the distributive property.

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

Step 2.1.2.5

Multiply $2$ by $- 1$ .

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

Step 2.1.2.6

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

$- x^{2} - 2 h x - h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $- x^{2}$ .

$- 2 h x - h^{2} - x^{2}$

Step 2.2.2

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

$- h^{2} - 2 h x - x^{2}$

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.2

Multiply $x^{2}$ by $1$ .

$\frac{- h^{2} - 2 h x - x^{2} + x^{2}}{h}$

Step 4.1.3

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

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Step 4.1.3.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 h x = 2 \cdot h \cdot x$

Step 4.1.3.2

Rewrite the polynomial.

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

Step 4.1.3.3

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = h$ and $b = x$ .

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

Step 4.1.4

Reorder $- {(h + x)}^{2}$ and $x^{2}$ .

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

Step 4.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = h + x$ .

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

Step 4.1.6

Simplify.

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Step 4.1.6.1

Add $x$ and $x$ .

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

Step 4.1.6.2

Apply the distributive property.

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

Step 4.1.6.3

Subtract $x$ from $x$ .

$\frac{(2 x + h) (- h + 0)}{h}$

Step 4.1.6.4

Add $- h$ and $0$ .

$\frac{(2 x + h) \cdot - 1 h}{h}$

Step 4.1.6.5

Factor out negative.

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

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

Step 4.2

Simplify terms.

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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.

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

Step 4.2.1.2

Divide $- (2 x + h)$ by $1$ .

$- (2 x + h)$

Step 4.2.2

Apply the distributive property.

$- (2 x) - h$

Step 4.2.3

Multiply $2$ by $- 1$ .

$- 2 x - h$

Step 5