Calculus Examples

$y = 5 + 0.6 x - 0.3 x^{2}$

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.

$f (x + h) = 5 + 0.6 (x + h) - 0.3 {(x + h)}^{2}$

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

Apply the distributive property.

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 {(x + h)}^{2}$

Step 2.1.2.1.2

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

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 ((x + h) (x + h))$

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.

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 (x (x + h) + h (x + h))$

Step 2.1.2.1.3.2

Apply the distributive property.

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 (x \cdot x + x h + h (x + h))$

Step 2.1.2.1.3.3

Apply the distributive property.

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 (x \cdot x + x h + h x + h \cdot h)$

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 $x$ .

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 (x^{2} + x h + h x + h \cdot h)$

Step 2.1.2.1.4.1.2

Multiply $h$ by $h$ .

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 (x^{2} + x h + h x + h^{2})$

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 $h$ .

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 (x^{2} + h x + h x + h^{2})$

Step 2.1.2.1.4.2.2

Add $h x$ and $h x$ .

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 (x^{2} + 2 h x + h^{2})$

Step 2.1.2.1.5

Apply the distributive property.

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 x^{2} - 0.3 (2 h x) - 0.3 h^{2}$

Step 2.1.2.1.6

Multiply $2$ by $- 0.3$ .

$f (x + h) = 5 + 0.6 x + 0.6 h - 0.3 x^{2} - 0.6 h x - 0.3 h^{2}$

Step 2.1.2.2

The final answer is $5 + 0.6 x + 0.6 h - 0.3 x^{2} - 0.6 h x - 0.3 h^{2}$ .

$5 + 0.6 x + 0.6 h - 0.3 x^{2} - 0.6 h x - 0.3 h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $5$ .

$0.6 x + 0.6 h - 0.3 x^{2} - 0.6 h x - 0.3 h^{2} + 5$

Step 2.2.2

Move $0.6 x$ .

$0.6 h - 0.3 x^{2} - 0.6 h x - 0.3 h^{2} + 0.6 x + 5$

Step 2.2.3

Move $0.6 h$ .

$- 0.3 x^{2} - 0.6 h x - 0.3 h^{2} + 0.6 h + 0.6 x + 5$

Step 2.2.4

Move $- 0.3 x^{2}$ .

$- 0.6 h x - 0.3 h^{2} - 0.3 x^{2} + 0.6 h + 0.6 x + 5$

Step 2.2.5

Reorder $- 0.6 h x$ and $- 0.3 h^{2}$ .

$- 0.3 h^{2} - 0.6 h x - 0.3 x^{2} + 0.6 h + 0.6 x + 5$

Step 2.3

Find the components of the definition.

$f (x + h) = - 0.3 h^{2} - 0.6 h x - 0.3 x^{2} + 0.6 h + 0.6 x + 5$

$f (x) = 5 + 0.6 x - 0.3 x^{2}$

$f (x + h) = - 0.3 h^{2} - 0.6 h x - 0.3 x^{2} + 0.6 h + 0.6 x + 5$

$f (x) = 5 + 0.6 x - 0.3 x^{2}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{- 0.3 h^{2} - 0.6 h x - 0.3 x^{2} + 0.6 h + 0.6 x + 5 - (5 + 0.6 x - 0.3 x^{2})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $0.1$ out of $- 0.3 h^{2} - 0.6 h x - 0.3 x^{2} + 0.6 h + 0.6 x + 5 - (5 + 0.6 x - 0.3 x^{2})$ .

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Step 4.1.1.1

Reorder the expression.

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Step 4.1.1.1.1

Move $h$ .

$f' (x) = lim_{h \to 0} \frac{- 0.3 h^{2} - 0.6 x h - 0.3 x^{2} + 0.6 h + 0.6 x + 5 - (5 + 0.6 x - 0.3 x^{2})}{h}$

Step 4.1.1.1.2

Move $5$ .

$f' (x) = lim_{h \to 0} \frac{- 0.3 h^{2} - 0.6 x h - 0.3 x^{2} + 0.6 h + 0.6 x + 5 - (0.6 x - 0.3 x^{2} + 5)}{h}$

Step 4.1.1.1.3

Reorder $0.6 x$ and $- 0.3 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{- 0.3 h^{2} - 0.6 x h - 0.3 x^{2} + 0.6 h + 0.6 x + 5 - (- 0.3 x^{2} + 0.6 x + 5)}{h}$

Step 4.1.1.2

Factor $0.1$ out of $- 0.3 h^{2}$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2}) - 0.6 x h - 0.3 x^{2} + 0.6 h + 0.6 x + 5 - (- 0.3 x^{2} + 0.6 x + 5)}{h}$

Step 4.1.1.3

Factor $0.1$ out of $- 0.6 x h$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2}) + 0.1 (- 6 x h) - 0.3 x^{2} + 0.6 h + 0.6 x + 5 - (- 0.3 x^{2} + 0.6 x + 5)}{h}$

Step 4.1.1.4

Factor $0.1$ out of $- 0.3 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2}) + 0.1 (- 6 x h) + 0.1 (- 3 x^{2}) + 0.6 h + 0.6 x + 5 - (- 0.3 x^{2} + 0.6 x + 5)}{h}$

Step 4.1.1.5

Factor $0.1$ out of $0.6 h$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2}) + 0.1 (- 6 x h) + 0.1 (- 3 x^{2}) + 0.1 (6 h) + 0.6 x + 5 - (- 0.3 x^{2} + 0.6 x + 5)}{h}$

Step 4.1.1.6

Factor $0.1$ out of $0.6 x$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2}) + 0.1 (- 6 x h) + 0.1 (- 3 x^{2}) + 0.1 (6 h) + 0.1 (6 x) + 5 - (- 0.3 x^{2} + 0.6 x + 5)}{h}$

Step 4.1.1.7

Factor $0.1$ out of $5$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2}) + 0.1 (- 6 x h) + 0.1 (- 3 x^{2}) + 0.1 (6 h) + 0.1 (6 x) + 0.1 (50) - (- 0.3 x^{2} + 0.6 x + 5)}{h}$

Step 4.1.1.8

Factor $0.1$ out of $- (- 0.3 x^{2} + 0.6 x + 5)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2}) + 0.1 (- 6 x h) + 0.1 (- 3 x^{2}) + 0.1 (6 h) + 0.1 (6 x) + 0.1 (50) + 0.1 (- 10 (- 0.3 x^{2} + 0.6 x + 5))}{h}$

Step 4.1.1.9

Factor $0.1$ out of $0.1 (- 3 h^{2}) + 0.1 (- 6 x h)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h) + 0.1 (- 3 x^{2}) + 0.1 (6 h) + 0.1 (6 x) + 0.1 (50) + 0.1 (- 10 (- 0.3 x^{2} + 0.6 x + 5))}{h}$

Step 4.1.1.10

Factor $0.1$ out of $0.1 (- 3 h^{2} - 6 x h) + 0.1 (- 3 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2}) + 0.1 (6 h) + 0.1 (6 x) + 0.1 (50) + 0.1 (- 10 (- 0.3 x^{2} + 0.6 x + 5))}{h}$

Step 4.1.1.11

Factor $0.1$ out of $0.1 (- 3 h^{2} - 6 x h - 3 x^{2}) + 0.1 (6 h)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h) + 0.1 (6 x) + 0.1 (50) + 0.1 (- 10 (- 0.3 x^{2} + 0.6 x + 5))}{h}$

Step 4.1.1.12

Factor $0.1$ out of $0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h) + 0.1 (6 x)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x) + 0.1 (50) + 0.1 (- 10 (- 0.3 x^{2} + 0.6 x + 5))}{h}$

Step 4.1.1.13

Factor $0.1$ out of $0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x) + 0.1 (50)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x + 50) + 0.1 (- 10 (- 0.3 x^{2} + 0.6 x + 5))}{h}$

Step 4.1.1.14

Factor $0.1$ out of $0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x + 50) + 0.1 (- 10 (- 0.3 x^{2} + 0.6 x + 5))$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x + 50 - 10 (- 0.3 x^{2} + 0.6 x + 5))}{h}$

Step 4.1.2

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x + 50 - 10 (- 0.3 x^{2}) - 10 (0.6 x) - 10 \cdot 5)}{h}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $- 0.3$ by $- 10$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x + 50 + 3 x^{2} - 10 (0.6 x) - 10 \cdot 5)}{h}$

Step 4.1.3.2

Multiply $0.6$ by $- 10$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x + 50 + 3 x^{2} - 6 x - 10 \cdot 5)}{h}$

Step 4.1.3.3

Multiply $- 10$ by $5$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h - 3 x^{2} + 6 h + 6 x + 50 + 3 x^{2} - 6 x - 50)}{h}$

Step 4.1.4

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

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h + 0 x^{2} + 6 h + 6 x + 50 - 6 x - 50)}{h}$

Step 4.1.5

Subtract $6 x$ from $6 x$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h + 0 x^{2} + 6 h + 0 x + 50 - 50)}{h}$

Step 4.1.6

Subtract $50$ from $50$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h + 0 x^{2} + 6 h + 0 x + 0)}{h}$

Step 4.1.7

Add $- 3 h^{2} - 6 x h + 0 x^{2} + 6 h + 0 x$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (- 3 h^{2} - 6 x h + 0 x^{2} + 6 h + 0 x)}{h}$

Step 4.1.8

Rewrite $- 3 h^{2} - 6 x h + 0 x^{2} + 6 h + 0 x$ in a factored form.

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Step 4.1.8.1

Factor $3$ out of $- 3 h^{2} - 6 x h + 0 x^{2} + 6 h + 0 x$ .

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Step 4.1.8.1.1

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

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2}) - 6 x h + 0 x^{2} + 6 h + 0 x)}{h}$

Step 4.1.8.1.2

Factor $3$ out of $- 6 x h$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2}) + 3 (- 2 x h) + 0 x^{2} + 6 h + 0 x)}{h}$

Step 4.1.8.1.3

Factor $3$ out of $0 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2}) + 3 (- 2 x h) + 3 (0 x^{2}) + 6 h + 0 x)}{h}$

Step 4.1.8.1.4

Factor $3$ out of $6 h$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2}) + 3 (- 2 x h) + 3 (0 x^{2}) + 3 (2 h) + 0 x)}{h}$

Step 4.1.8.1.5

Factor $3$ out of $0 x$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2}) + 3 (- 2 x h) + 3 (0 x^{2}) + 3 (2 h) + 3 (0 x))}{h}$

Step 4.1.8.1.6

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

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h) + 3 (0 x^{2}) + 3 (2 h) + 3 (0 x))}{h}$

Step 4.1.8.1.7

Factor $3$ out of $3 (- h^{2} - 2 x h) + 3 (0 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h + 0 x^{2}) + 3 (2 h) + 3 (0 x))}{h}$

Step 4.1.8.1.8

Factor $3$ out of $3 (- h^{2} - 2 x h + 0 x^{2}) + 3 (2 h)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h + 0 x^{2} + 2 h) + 3 (0 x))}{h}$

Step 4.1.8.1.9

Factor $3$ out of $3 (- h^{2} - 2 x h + 0 x^{2} + 2 h) + 3 (0 x)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h + 0 x^{2} + 2 h + 0 x))}{h}$

Step 4.1.8.2

Multiply $0$ by $x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h + 0 + 2 h + 0 x))}{h}$

Step 4.1.8.3

Multiply $0$ by $x$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h + 0 + 2 h + 0))}{h}$

Step 4.1.8.4

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

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h + 0 + 2 h))}{h}$

Step 4.1.8.5

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

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (- h^{2} - 2 x h + 2 h))}{h}$

Step 4.1.8.6

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

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Step 4.1.8.6.1

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

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (h (- h) - 2 x h + 2 h))}{h}$

Step 4.1.8.6.2

Factor $h$ out of $- 2 x h$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (h (- h) + h (- 2 x) + 2 h))}{h}$

Step 4.1.8.6.3

Factor $h$ out of $2 h$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (h (- h) + h (- 2 x) + h \cdot 2))}{h}$

Step 4.1.8.6.4

Factor $h$ out of $h (- h) + h (- 2 x)$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (h (- h - 2 x) + h \cdot 2))}{h}$

Step 4.1.8.6.5

Factor $h$ out of $h (- h - 2 x) + h \cdot 2$ .

$f' (x) = lim_{h \to 0} \frac{0.1 (3 (h (- h - 2 x + 2)))}{h}$

$f' (x) = lim_{h \to 0} \frac{0.1 (3 h (- h - 2 x + 2))}{h}$

$f' (x) = lim_{h \to 0} \frac{0.1 \cdot (3 h (- h - 2 x + 2))}{h}$

Step 4.1.9

Multiply $0.1$ by $3$ .

$f' (x) = lim_{h \to 0} \frac{0.3 h (- h - 2 x + 2)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{0.3 h (- h - 2 x + 2)}{h}$

Step 4.2.1.2

Divide $0.3 (- h - 2 x + 2)$ by $1$ .

$f' (x) = lim_{h \to 0} 0.3 (- h - 2 x + 2)$

Step 4.2.2

Apply the distributive property.

$f' (x) = lim_{h \to 0} 0.3 (- h) + 0.3 (- 2 x) + 0.3 \cdot 2$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $- 1$ by $0.3$ .

$f' (x) = lim_{h \to 0} - 0.3 h + 0.3 (- 2 x) + 0.3 \cdot 2$

Step 4.3.2

Multiply $- 2$ by $0.3$ .

$f' (x) = lim_{h \to 0} - 0.3 h - 0.6 x + 0.3 \cdot 2$

Step 4.3.3

Multiply $0.3$ by $2$ .

$f' (x) = lim_{h \to 0} - 0.3 h - 0.6 x + 0.6$

Step 4.4

Reorder $- 0.3 h$ and $- 0.6 x$ .

$f' (x) = lim_{h \to 0} - 0.6 x - 0.3 h + 0.6$

Step 5

Evaluate the limit.

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Step 5.1

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

$- lim_{h \to 0} 0.6 x - lim_{h \to 0} 0.3 h + lim_{h \to 0} 0.6$

Step 5.2

Evaluate the limit of $0.6 x$ which is constant as $h$ approaches $0$ .

$- 0.6 x - lim_{h \to 0} 0.3 h + lim_{h \to 0} 0.6$

Step 5.3

Move the term $0.3$ outside of the limit because it is constant with respect to $h$ .

$- 0.6 x - 0.3 lim_{h \to 0} h + lim_{h \to 0} 0.6$

Step 5.4

Evaluate the limit of $0.6$ which is constant as $h$ approaches $0$ .

$- 0.6 x - 0.3 lim_{h \to 0} h + 0.6$

Step 6

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

$- 0.6 x - 0.3 \cdot 0 + 0.6$

Step 7

Simplify the answer.

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Step 7.1

Multiply $- 0.3$ by $0$ .

$- 0.6 x + 0 + 0.6$

Step 7.2

Add $- 0.6 x$ and $0$ .

$- 0.6 x + 0.6$

Step 8