Calculus Examples

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

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

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.

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

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.

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

Step 2.1.2.1.2

Apply the distributive property.

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

Step 2.1.2.2

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

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

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

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

Step 2.2.2

Move $x$ .

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

Step 2.2.3

Move $- 3 x$ .

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

Step 2.2.4

Move $x^{3}$ .

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

Step 2.2.5

Move $3 h x^{2}$ .

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

Step 2.2.6

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

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 3 h - 3 x - (x^{3} - 3 x)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 3 h - 3 x - x^{3} - (- 3 x)}{h}$

Step 4.1.2

Multiply $- 3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 3 h - 3 x - x^{3} + 3 x}{h}$

Step 4.1.3

Subtract $x^{3}$ from $x^{3}$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 3 h - 3 x + 0 + 3 x}{h}$

Step 4.1.4

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

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 3 h - 3 x + 3 x}{h}$

Step 4.1.5

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

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 3 h + 0}{h}$

Step 4.1.6

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

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 3 h}{h}$

Step 4.1.7

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

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Step 4.1.7.1

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

$f' (x) = lim_{h \to 0} \frac{h \cdot h^{2} + 3 h^{2} x + 3 h x^{2} - 3 h}{h}$

Step 4.1.7.2

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + 3 h x^{2} - 3 h}{h}$

Step 4.1.7.3

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) - 3 h}{h}$

Step 4.1.7.4

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h \cdot - 3}{h}$

Step 4.1.7.5

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x) + h (3 x^{2}) + h \cdot - 3}{h}$

Step 4.1.7.6

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2}) + h \cdot - 3}{h}$

Step 4.1.7.7

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 3)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 3)}{h}$

Step 4.2.1.2

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

$f' (x) = lim_{h \to 0} h^{2} + 3 h x + 3 x^{2} - 3$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

$f' (x) = lim_{h \to 0} h^{2} + 3 x h + 3 x^{2} - 3$

Step 4.2.2.2

Move $h^{2}$ .

$f' (x) = lim_{h \to 0} 3 x h + 3 x^{2} + h^{2} - 3$

Step 4.2.2.3

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

$f' (x) = lim_{h \to 0} 3 x^{2} + 3 x h + h^{2} - 3$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

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

Step 6

Evaluate the limit of $3 x^{2}$ which is constant as $h$ approaches $0$ .

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

Step 7

Move the term $3 x$ outside of the limit because it is constant with respect to $h$ .

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

Step 8

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$3 x^{2} + 3 x lim_{h \to 0} h + {(lim_{h \to 0} h)}^{2} - lim_{h \to 0} 3$

Step 9

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$3 x^{2} + 3 x lim_{h \to 0} h + {(lim_{h \to 0} h)}^{2} - 1 \cdot 3$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 10.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$3 x^{2} + 3 x \cdot 0 + {(lim_{h \to 0} h)}^{2} - 1 \cdot 3$

Step 10.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

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

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $3 x \cdot 0$ .

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Step 11.1.1.1

Multiply $0$ by $3$ .

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

Step 11.1.1.2

Multiply $0$ by $x$ .

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

Step 11.1.2

Raising $0$ to any positive power yields $0$ .

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

Step 11.1.3

Multiply $- 1$ by $3$ .

$3 x^{2} + 0 + 0 - 3$

Step 11.2

Combine the opposite terms in $3 x^{2} + 0 + 0 - 3$ .

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Step 11.2.1

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

$3 x^{2} + 0 - 3$

Step 11.2.2

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

$3 x^{2} - 3$

Step 12