Calculus Examples

$f (x) = x^{3} - 5 x^{2} + 6$

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.

$f (x + h) = {(x + h)}^{3} - 5 {(x + h)}^{2} + 6$

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.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 {(x + h)}^{2} + 6$

Step 2.1.2.1.2

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

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 ((x + h) (x + h)) + 6$

Step 2.1.2.1.3

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

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Step 2.1.2.1.3.1

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 (x (x + h) + h (x + h)) + 6$

Step 2.1.2.1.3.2

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 (x \cdot x + x h + h (x + h)) + 6$

Step 2.1.2.1.3.3

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 (x \cdot x + x h + h x + h \cdot h) + 6$

Step 2.1.2.1.4

Simplify and combine like terms.

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Step 2.1.2.1.4.1

Simplify each term.

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Step 2.1.2.1.4.1.1

Multiply $x$ by $x$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 (x^{2} + x h + h x + h \cdot h) + 6$

Step 2.1.2.1.4.1.2

Multiply $h$ by $h$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 (x^{2} + x h + h x + h^{2}) + 6$

Step 2.1.2.1.4.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.4.2.1

Reorder $x$ and $h$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 (x^{2} + h x + h x + h^{2}) + 6$

Step 2.1.2.1.4.2.2

Add $h x$ and $h x$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 (x^{2} + 2 h x + h^{2}) + 6$

Step 2.1.2.1.5

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 x^{2} - 5 (2 h x) - 5 h^{2} + 6$

Step 2.1.2.1.6

Multiply $2$ by $- 5$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 x^{2} - 10 h x - 5 h^{2} + 6$

Step 2.1.2.2

The final answer is $x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 x^{2} - 10 h x - 5 h^{2} + 6$ .

$x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 5 x^{2} - 10 h x - 5 h^{2} + 6$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$x^{3} + 3 h x^{2} + 3 x h^{2} + h^{3} - 5 x^{2} - 10 h x - 5 h^{2} + 6$

Step 2.2.2

Move $x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 5 x^{2} - 10 h x - 5 h^{2} + 6$

Step 2.2.3

Move $- 5 x^{2}$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 10 h x - 5 h^{2} - 5 x^{2} + 6$

Step 2.2.4

Move $- 10 h x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6$

Step 2.2.5

Move $x^{3}$ .

$3 h x^{2} + 3 h^{2} x + h^{3} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6$

Step 2.2.6

Move $3 h x^{2}$ .

$3 h^{2} x + h^{3} + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6$

Step 2.2.7

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

$h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6$

$f (x) = x^{3} - 5 x^{2} + 6$

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6$

$f (x) = x^{3} - 5 x^{2} + 6$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6 - (x^{3} - 5 x^{2} + 6)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6 - x^{3} - (- 5 x^{2}) - 1 \cdot 6}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- 5$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6 - x^{3} + 5 x^{2} - 1 \cdot 6}{h}$

Step 4.1.2.2

Multiply $- 1$ by $6$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 5 h^{2} - 10 h x - 5 x^{2} + 6 - x^{3} + 5 x^{2} - 6}{h}$

Step 4.1.3

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

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x - 5 x^{2} + 6 + 0 + 5 x^{2} - 6}{h}$

Step 4.1.4

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

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x - 5 x^{2} + 6 + 5 x^{2} - 6}{h}$

Step 4.1.5

Add $- 5 x^{2}$ and $5 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x + 0 + 6 - 6}{h}$

Step 4.1.6

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

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x + 6 - 6}{h}$

Step 4.1.7

Subtract $6$ from $6$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x + 0}{h}$

Step 4.1.8

Add $h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x}{h}$

Step 4.1.9

Factor $h$ out of $h^{3} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x$ .

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Step 4.1.9.1

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

$f' (x) = lim_{h \to 0} \frac{h \cdot h^{2} + 3 h^{2} x + 3 h x^{2} - 5 h^{2} - 10 h x}{h}$

Step 4.1.9.2

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + 3 h x^{2} - 5 h^{2} - 10 h x}{h}$

Step 4.1.9.3

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) - 5 h^{2} - 10 h x}{h}$

Step 4.1.9.4

Factor $h$ out of $- 5 h^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h (- 5 h) - 10 h x}{h}$

Step 4.1.9.5

Factor $h$ out of $- 10 h x$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h (- 5 h) + h (- 10 x)}{h}$

Step 4.1.9.6

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x) + h (3 x^{2}) + h (- 5 h) + h (- 10 x)}{h}$

Step 4.1.9.7

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

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2}) + h (- 5 h) + h (- 10 x)}{h}$

Step 4.1.9.8

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2}) + h (- 5 h)$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 5 h) + h (- 10 x)}{h}$

Step 4.1.9.9

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2} - 5 h) + h (- 10 x)$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 5 h - 10 x)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 5 h - 10 x)}{h}$

Step 4.2.1.2

Divide $h^{2} + 3 h x + 3 x^{2} - 5 h - 10 x$ by $1$ .

$f' (x) = lim_{h \to 0} h^{2} + 3 h x + 3 x^{2} - 5 h - 10 x$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

$f' (x) = lim_{h \to 0} h^{2} + 3 x h + 3 x^{2} - 5 h - 10 x$

Step 4.2.2.2

Move $- 5 h$ .

$f' (x) = lim_{h \to 0} h^{2} + 3 x h + 3 x^{2} - 10 x - 5 h$

Step 4.2.2.3

Move $h^{2}$ .

$f' (x) = lim_{h \to 0} 3 x h + 3 x^{2} + h^{2} - 10 x - 5 h$

Step 4.2.2.4

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

$f' (x) = lim_{h \to 0} 3 x^{2} + 3 x h + h^{2} - 10 x - 5 h$

Step 5

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

$lim_{h \to 0} 3 x^{2} + lim_{h \to 0} 3 x h + lim_{h \to 0} h^{2} - lim_{h \to 0} 10 x - lim_{h \to 0} 5 h$

Step 6

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

$3 x^{2} + lim_{h \to 0} 3 x h + lim_{h \to 0} h^{2} - lim_{h \to 0} 10 x - lim_{h \to 0} 5 h$

Step 7

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

$3 x^{2} + 3 x lim_{h \to 0} h + lim_{h \to 0} h^{2} - lim_{h \to 0} 10 x - lim_{h \to 0} 5 h$

Step 8

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

$3 x^{2} + 3 x lim_{h \to 0} h + {(lim_{h \to 0} h)}^{2} - lim_{h \to 0} 10 x - lim_{h \to 0} 5 h$

Step 9

Evaluate the limit of $10 x$ which is constant as $h$ approaches $0$ .

$3 x^{2} + 3 x lim_{h \to 0} h + {(lim_{h \to 0} h)}^{2} - 10 x - lim_{h \to 0} 5 h$

Step 10

Move the term $5$ outside of the limit because it is constant with respect to $h$ .

$3 x^{2} + 3 x lim_{h \to 0} h + {(lim_{h \to 0} h)}^{2} - 10 x - 5 lim_{h \to 0} h$

Step 11

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

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Step 11.1

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

$3 x^{2} + 3 x \cdot 0 + {(lim_{h \to 0} h)}^{2} - 10 x - 5 lim_{h \to 0} h$

Step 11.2

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

$3 x^{2} + 3 x \cdot 0 + 0^{2} - 10 x - 5 lim_{h \to 0} h$

Step 11.3

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

$3 x^{2} + 3 x \cdot 0 + 0^{2} - 10 x - 5 \cdot 0$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply $3 x \cdot 0$ .

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Step 12.1.1.1

Multiply $0$ by $3$ .

$3 x^{2} + 0 x + 0^{2} - 10 x - 5 \cdot 0$

Step 12.1.1.2

Multiply $0$ by $x$ .

$3 x^{2} + 0 + 0^{2} - 10 x - 5 \cdot 0$

Step 12.1.2

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

$3 x^{2} + 0 + 0 - 10 x - 5 \cdot 0$

Step 12.1.3

Multiply $- 5$ by $0$ .

$3 x^{2} + 0 + 0 - 10 x + 0$

Step 12.2

Combine the opposite terms in $3 x^{2} + 0 + 0 - 10 x + 0$ .

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Step 12.2.1

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

$3 x^{2} + 0 - 10 x + 0$

Step 12.2.2

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

$3 x^{2} - 10 x + 0$

Step 12.2.3

Add $3 x^{2} - 10 x$ and $0$ .

$3 x^{2} - 10 x$

Step 13