Calculus Examples

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

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.

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

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 $(x + h) (x + h)$ .

$f (x + h) = (x + h) (x + h) - 1$

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.

$f (x + h) = x (x + h) + h (x + h) - 1$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) - 1$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h - 1$

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 $x$ .

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

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

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

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 $h$ .

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

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.2

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

$x^{2} + 2 h x + h^{2} - 1$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} - 1$

Step 2.2.2

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

$h^{2} + 2 h x + x^{2} - 1$

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Multiply $- 1$ by $- 1$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Add $- 1$ and $1$ .

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

Step 4.1.6

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

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

Step 4.1.7

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

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Step 4.1.7.1

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

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

Step 4.1.7.2

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

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

Step 4.1.7.3

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

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 $0$ .

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

Step 5.2

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

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

Step 6

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

$2 x + 0$

Step 7

Add $2 x$ and $0$ .

$2 x$

Step 8