Calculus Examples

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

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.

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

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

Use the Binomial Theorem.

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

Step 2.1.2.1.2

Apply the distributive property.

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

Step 2.1.2.1.3

Simplify.

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Step 2.1.2.1.3.1

Multiply $3$ by $- 1$ .

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

Step 2.1.2.1.3.2

Multiply $3$ by $- 1$ .

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

Step 2.1.2.1.4

Remove parentheses.

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

Step 2.1.2.2

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

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

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

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

Step 2.2.2

Move $x$ .

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

Step 2.2.3

Move $1$ .

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

Step 2.2.4

Move $- x^{3}$ .

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

Step 2.2.5

Move $- 3 h x^{2}$ .

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

Step 2.2.6

Reorder $- 3 h^{2} x$ and $- h^{3}$ .

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Multiply $- 1$ by $1$ .

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

Step 4.1.3

Multiply $- - x^{3}$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.3.2

Multiply $x^{3}$ by $1$ .

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

Step 4.1.4

Add $- x^{3}$ and $x^{3}$ .

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

Step 4.1.5

Add $- h^{3}$ and $0$ .

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

Step 4.1.6

Subtract $1$ from $1$ .

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

Step 4.1.7

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

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

Step 4.1.8

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

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Step 4.1.8.1

Factor $h$ out of $- h^{3}$ .

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

Step 4.1.8.2

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

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

Step 4.1.8.3

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

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

Step 4.1.8.4

Factor $h$ out of $h (- h^{2}) + h (- 3 h x)$ .

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

Step 4.1.8.5

Factor $h$ out of $h (- h^{2} - 3 h x) + h (- 3 x^{2})$ .

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

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.

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

Step 4.2.1.2

Divide $- h^{2} - 3 h x - 3 x^{2}$ by $1$ .

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

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

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

Step 4.2.2.2

Move $- h^{2}$ .

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

Step 4.2.2.3

Reorder $- 3 x h$ and $- 3 x^{2}$ .

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

Step 5