Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Use the Binomial Theorem.

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

Step 2.1.2.2.2

Apply the distributive property.

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

Step 2.1.2.2.3

Simplify.

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Step 2.1.2.2.3.1

Multiply $3$ by $4$ .

$f (x + h) = 4 x^{3} + 12 (x^{2} h) + 4 (3 x h^{2}) + 4 h^{3} + 2 {(x + h)}^{2} + x + h - 1$

Step 2.1.2.2.3.2

Multiply $3$ by $4$ .

$f (x + h) = 4 x^{3} + 12 (x^{2} h) + 12 (x h^{2}) + 4 h^{3} + 2 {(x + h)}^{2} + x + h - 1$

Step 2.1.2.2.4

Remove parentheses.

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 {(x + h)}^{2} + x + h - 1$

Step 2.1.2.2.5

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

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 ((x + h) (x + h)) + x + h - 1$

Step 2.1.2.2.6

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

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Step 2.1.2.2.6.1

Apply the distributive property.

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 (x (x + h) + h (x + h)) + x + h - 1$

Step 2.1.2.2.6.2

Apply the distributive property.

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 (x \cdot x + x h + h (x + h)) + x + h - 1$

Step 2.1.2.2.6.3

Apply the distributive property.

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 (x \cdot x + x h + h x + h \cdot h) + x + h - 1$

Step 2.1.2.2.7

Simplify and combine like terms.

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Step 2.1.2.2.7.1

Simplify each term.

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Step 2.1.2.2.7.1.1

Multiply $x$ by $x$ .

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 (x^{2} + x h + h x + h \cdot h) + x + h - 1$

Step 2.1.2.2.7.1.2

Multiply $h$ by $h$ .

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 (x^{2} + x h + h x + h^{2}) + x + h - 1$

Step 2.1.2.2.7.2

Add $x h$ and $h x$ .

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Step 2.1.2.2.7.2.1

Reorder $x$ and $h$ .

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 (x^{2} + h x + h x + h^{2}) + x + h - 1$

Step 2.1.2.2.7.2.2

Add $h x$ and $h x$ .

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 (x^{2} + 2 h x + h^{2}) + x + h - 1$

Step 2.1.2.2.8

Apply the distributive property.

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 x^{2} + 2 (2 h x) + 2 h^{2} + x + h - 1$

Step 2.1.2.2.9

Multiply $2$ by $2$ .

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 x^{2} + 4 h x + 2 h^{2} + x + h - 1$

Step 2.1.2.3

The final answer is $4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 x^{2} + 4 h x + 2 h^{2} + x + h - 1$ .

$4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} + 2 x^{2} + 4 h x + 2 h^{2} + x + h - 1$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$4 x^{3} + 12 h x^{2} + 12 x h^{2} + 4 h^{3} + 2 x^{2} + 4 h x + 2 h^{2} + x + h - 1$

Step 2.2.2

Move $x$ .

$4 x^{3} + 12 h x^{2} + 12 h^{2} x + 4 h^{3} + 2 x^{2} + 4 h x + 2 h^{2} + x + h - 1$

Step 2.2.3

Move $x$ .

$4 x^{3} + 12 h x^{2} + 12 h^{2} x + 4 h^{3} + 2 x^{2} + 4 h x + 2 h^{2} + h + x - 1$

Step 2.2.4

Move $2 x^{2}$ .

$4 x^{3} + 12 h x^{2} + 12 h^{2} x + 4 h^{3} + 4 h x + 2 h^{2} + 2 x^{2} + h + x - 1$

Step 2.2.5

Move $4 h x$ .

$4 x^{3} + 12 h x^{2} + 12 h^{2} x + 4 h^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1$

Step 2.2.6

Move $4 x^{3}$ .

$12 h x^{2} + 12 h^{2} x + 4 h^{3} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1$

Step 2.2.7

Move $12 h x^{2}$ .

$12 h^{2} x + 4 h^{3} + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1$

Step 2.2.8

Reorder $12 h^{2} x$ and $4 h^{3}$ .

$4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1$

Step 2.3

Find the components of the definition.

$f (x + h) = 4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1$

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

$f (x + h) = 4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1$

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

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1 - (4 x^{3} + 2 x^{2} + x - 1)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1 - (4 x^{3}) - (2 x^{2}) - x + 1}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $4$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1 - 4 x^{3} - (2 x^{2}) - x + 1}{h}$

Step 4.1.2.2

Multiply $2$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1 - 4 x^{3} - 2 x^{2} - x + 1}{h}$

Step 4.1.2.3

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1 - 4 x^{3} - 2 x^{2} - x + 1}{h}$

Step 4.1.3

Subtract $4 x^{3}$ from $4 x^{3}$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1 + 0 - 2 x^{2} - x + 1}{h}$

Step 4.1.4

Add $4 h^{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + 2 x^{2} + h + x - 1 - 2 x^{2} - x + 1}{h}$

Step 4.1.5

Subtract $2 x^{2}$ from $2 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h + x - 1 + 0 - x + 1}{h}$

Step 4.1.6

Add $4 h^{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h + x - 1 - x + 1}{h}$

Step 4.1.7

Subtract $x$ from $x$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h + 0 - 1 + 1}{h}$

Step 4.1.8

Add $4 h^{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h - 1 + 1}{h}$

Step 4.1.9

Add $- 1$ and $1$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h + 0}{h}$

Step 4.1.10

Add $4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h}{h}$

Step 4.1.11

Factor $h$ out of $4 h^{3} + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h$ .

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Step 4.1.11.1

Factor $h$ out of $4 h^{3}$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2}) + 12 h^{2} x + 12 h x^{2} + 2 h^{2} + 4 h x + h}{h}$

Step 4.1.11.2

Factor $h$ out of $12 h^{2} x$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2}) + h (12 h x) + 12 h x^{2} + 2 h^{2} + 4 h x + h}{h}$

Step 4.1.11.3

Factor $h$ out of $12 h x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2}) + h (12 h x) + h (12 x^{2}) + 2 h^{2} + 4 h x + h}{h}$

Step 4.1.11.4

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

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2}) + h (12 h x) + h (12 x^{2}) + h (2 h) + 4 h x + h}{h}$

Step 4.1.11.5

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

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2}) + h (12 h x) + h (12 x^{2}) + h (2 h) + h (4 x) + h}{h}$

Step 4.1.11.6

Raise $h$ to the power of $1$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2}) + h (12 h x) + h (12 x^{2}) + h (2 h) + h (4 x) + h}{h}$

Step 4.1.11.7

Factor $h$ out of $h^{1}$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2}) + h (12 h x) + h (12 x^{2}) + h (2 h) + h (4 x) + h \cdot 1}{h}$

Step 4.1.11.8

Factor $h$ out of $h (4 h^{2}) + h (12 h x)$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2} + 12 h x) + h (12 x^{2}) + h (2 h) + h (4 x) + h \cdot 1}{h}$

Step 4.1.11.9

Factor $h$ out of $h (4 h^{2} + 12 h x) + h (12 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2} + 12 h x + 12 x^{2}) + h (2 h) + h (4 x) + h \cdot 1}{h}$

Step 4.1.11.10

Factor $h$ out of $h (4 h^{2} + 12 h x + 12 x^{2}) + h (2 h)$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2} + 12 h x + 12 x^{2} + 2 h) + h (4 x) + h \cdot 1}{h}$

Step 4.1.11.11

Factor $h$ out of $h (4 h^{2} + 12 h x + 12 x^{2} + 2 h) + h (4 x)$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2} + 12 h x + 12 x^{2} + 2 h + 4 x) + h \cdot 1}{h}$

Step 4.1.11.12

Factor $h$ out of $h (4 h^{2} + 12 h x + 12 x^{2} + 2 h + 4 x) + h \cdot 1$ .

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2} + 12 h x + 12 x^{2} + 2 h + 4 x + 1)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h (4 h^{2} + 12 h x + 12 x^{2} + 2 h + 4 x + 1)}{h}$

Step 4.2.1.2

Divide $4 h^{2} + 12 h x + 12 x^{2} + 2 h + 4 x + 1$ by $1$ .

$f' (x) = lim_{h \to 0} 4 h^{2} + 12 h x + 12 x^{2} + 2 h + 4 x + 1$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

$f' (x) = lim_{h \to 0} 4 h^{2} + 12 x h + 12 x^{2} + 2 h + 4 x + 1$

Step 4.2.2.2

Move $2 h$ .

$f' (x) = lim_{h \to 0} 4 h^{2} + 12 x h + 12 x^{2} + 4 x + 2 h + 1$

Step 4.2.2.3

Move $4 h^{2}$ .

$f' (x) = lim_{h \to 0} 12 x h + 12 x^{2} + 4 h^{2} + 4 x + 2 h + 1$

Step 4.2.2.4

Reorder $12 x h$ and $12 x^{2}$ .

$f' (x) = lim_{h \to 0} 12 x^{2} + 12 x h + 4 h^{2} + 4 x + 2 h + 1$

Step 5

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

$lim_{h \to 0} 12 x^{2} + lim_{h \to 0} 12 x h + lim_{h \to 0} 4 h^{2} + lim_{h \to 0} 4 x + lim_{h \to 0} 2 h + lim_{h \to 0} 1$

Step 6

Evaluate the limit of $12 x^{2}$ which is constant as $h$ approaches $0$ .

$12 x^{2} + lim_{h \to 0} 12 x h + lim_{h \to 0} 4 h^{2} + lim_{h \to 0} 4 x + lim_{h \to 0} 2 h + lim_{h \to 0} 1$

Step 7

Move the term $12 x$ outside of the limit because it is constant with respect to $h$ .

$12 x^{2} + 12 x lim_{h \to 0} h + lim_{h \to 0} 4 h^{2} + lim_{h \to 0} 4 x + lim_{h \to 0} 2 h + lim_{h \to 0} 1$

Step 8

Move the term $4$ outside of the limit because it is constant with respect to $h$ .

$12 x^{2} + 12 x lim_{h \to 0} h + 4 lim_{h \to 0} h^{2} + lim_{h \to 0} 4 x + lim_{h \to 0} 2 h + lim_{h \to 0} 1$

Step 9

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

$12 x^{2} + 12 x lim_{h \to 0} h + 4 {(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 4 x + lim_{h \to 0} 2 h + lim_{h \to 0} 1$

Step 10

Evaluate the limit of $4 x$ which is constant as $h$ approaches $0$ .

$12 x^{2} + 12 x lim_{h \to 0} h + 4 {(lim_{h \to 0} h)}^{2} + 4 x + lim_{h \to 0} 2 h + lim_{h \to 0} 1$

Step 11

Move the term $2$ outside of the limit because it is constant with respect to $h$ .

$12 x^{2} + 12 x lim_{h \to 0} h + 4 {(lim_{h \to 0} h)}^{2} + 4 x + 2 lim_{h \to 0} h + lim_{h \to 0} 1$

Step 12

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$12 x^{2} + 12 x lim_{h \to 0} h + 4 {(lim_{h \to 0} h)}^{2} + 4 x + 2 lim_{h \to 0} h + 1$

Step 13

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

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Step 13.1

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

$12 x^{2} + 12 x \cdot 0 + 4 {(lim_{h \to 0} h)}^{2} + 4 x + 2 lim_{h \to 0} h + 1$

Step 13.2

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

$12 x^{2} + 12 x \cdot 0 + 4 \cdot 0^{2} + 4 x + 2 lim_{h \to 0} h + 1$

Step 13.3

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

$12 x^{2} + 12 x \cdot 0 + 4 \cdot 0^{2} + 4 x + 2 \cdot 0 + 1$

Step 14

Simplify the answer.

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Step 14.1

Simplify each term.

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Step 14.1.1

Multiply $12 x \cdot 0$ .

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Step 14.1.1.1

Multiply $0$ by $12$ .

$12 x^{2} + 0 x + 4 \cdot 0^{2} + 4 x + 2 \cdot 0 + 1$

Step 14.1.1.2

Multiply $0$ by $x$ .

$12 x^{2} + 0 + 4 \cdot 0^{2} + 4 x + 2 \cdot 0 + 1$

Step 14.1.2

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

$12 x^{2} + 0 + 4 \cdot 0 + 4 x + 2 \cdot 0 + 1$

Step 14.1.3

Multiply $4$ by $0$ .

$12 x^{2} + 0 + 0 + 4 x + 2 \cdot 0 + 1$

Step 14.1.4

Multiply $2$ by $0$ .

$12 x^{2} + 0 + 0 + 4 x + 0 + 1$

Step 14.2

Combine the opposite terms in $12 x^{2} + 0 + 0 + 4 x + 0 + 1$ .

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Step 14.2.1

Add $12 x^{2}$ and $0$ .

$12 x^{2} + 0 + 4 x + 0 + 1$

Step 14.2.2

Add $12 x^{2}$ and $0$ .

$12 x^{2} + 4 x + 0 + 1$

Step 14.2.3

Add $12 x^{2} + 4 x$ and $0$ .

$12 x^{2} + 4 x + 1$

Step 15