Calculus Examples

$g (x) = 9 - x^{2}$

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.

$f (x + h) = 9 - {(x + h)}^{2}$

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

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

$f (x + h) = 9 - ((x + h) (x + h))$

Step 2.1.2.1.2

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

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = 9 - (x (x + h) + h (x + h))$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 9 - (x \cdot x + x h + h (x + h))$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 9 - (x \cdot x + x h + h x + h \cdot h)$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = 9 - (x^{2} + x h + h x + h \cdot h)$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = 9 - (x^{2} + x h + h x + h^{2})$

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = 9 - (x^{2} + h x + h x + h^{2})$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = 9 - (x^{2} + 2 h x + h^{2})$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = 9 - x^{2} - (2 h x) - h^{2}$

Step 2.1.2.1.5

Multiply $2$ by $- 1$ .

$f (x + h) = 9 - x^{2} - 2 h x - h^{2}$

Step 2.1.2.2

The final answer is $9 - x^{2} - 2 h x - h^{2}$ .

$9 - x^{2} - 2 h x - h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $9$ .

$- x^{2} - 2 h x - h^{2} + 9$

Step 2.2.2

Move $- x^{2}$ .

$- 2 h x - h^{2} - x^{2} + 9$

Step 2.2.3

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

$- h^{2} - 2 h x - x^{2} + 9$

Step 2.3

Find the components of the definition.

$f (x + h) = - h^{2} - 2 h x - x^{2} + 9$

$f (x) = 9 - x^{2}$

$f (x + h) = - h^{2} - 2 h x - x^{2} + 9$

$f (x) = 9 - x^{2}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + 9 - (9 - x^{2})}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + 9 - 1 \cdot 9 + x^{2}}{h}$

Step 4.1.2

Multiply $- 1$ by $9$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + 9 - 9 + x^{2}}{h}$

Step 4.1.3

Multiply $- - x^{2}$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + 9 - 9 + 1 x^{2}}{h}$

Step 4.1.3.2

Multiply $x^{2}$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x - x^{2} + 9 - 9 + x^{2}}{h}$

Step 4.1.4

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

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x + 0 + 9 - 9}{h}$

Step 4.1.5

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

$f' (x) = lim_{h \to 0} \frac{- h^{2} - 2 h x + 9 - 9}{h}$

Step 4.1.6

Subtract $9$ from $9$ .

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

Step 4.1.7

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

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

Step 4.1.8

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

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Step 4.1.8.1

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

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

Step 4.1.8.2

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

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

Step 4.1.8.3

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

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

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.

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

Step 4.2.1.2

Divide $- h - 2 x$ by $1$ .

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

Step 4.2.2

Reorder $- h$ and $- 2 x$ .

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

Step 5

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

$- lim_{h \to 0} 2 x - lim_{h \to 0} h$

Step 6

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

$- 2 x - lim_{h \to 0} h$

Step 7

Simplify the expression.

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Step 7.1

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

$- 2 x + 0$

Step 7.2

Add $- 2 x$ and $0$ .

$- 2 x$

Step 8