Calculus Examples

$f (x) = 3 x^{2}$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

$f (x + h) = 3 ((x + h) (x + h))$

Step 2.1.2.2

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

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Step 2.1.2.2.1

Apply the distributive property.

$f (x + h) = 3 (x (x + h) + h (x + h))$

Step 2.1.2.2.2

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h (x + h))$

Step 2.1.2.2.3

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h x + h \cdot h)$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.4

Apply the distributive property.

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

Step 2.1.2.5

Multiply $2$ by $3$ .

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

Step 2.1.2.6

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

$3 x^{2} + 6 h x + 3 h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $3 x^{2}$ .

$6 h x + 3 h^{2} + 3 x^{2}$

Step 2.2.2

Reorder $6 h x$ and $3 h^{2}$ .

$3 h^{2} + 6 h x + 3 x^{2}$

Step 2.3

Find the components of the definition.

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

$f (x) = 3 x^{2}$

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

$f (x) = 3 x^{2}$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $3$ .

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

Step 4.1.2

Subtract $3 x^{2}$ from $3 x^{2}$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

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Step 4.1.4.1

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

$f' (x) = lim_{h \to 0} \frac{3 h \cdot h + 6 h x}{h}$

Step 4.1.4.2

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

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

Step 4.1.4.3

Factor $3 h$ out of $3 h \cdot h + 3 h (2 x)$ .

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

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.

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

Step 4.2.1.2

Divide $3 (h + 2 x)$ by $1$ .

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply $2$ by $3$ .

$f' (x) = lim_{h \to 0} 3 h + 6 x$

Step 4.2.3.2

Reorder $3 h$ and $6 x$ .

$f' (x) = lim_{h \to 0} 6 x + 3 h$

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 $0$ .

$lim_{h \to 0} 6 x + lim_{h \to 0} 3 h$

Step 5.2

Evaluate the limit of $6 x$ which is constant as $h$ approaches $0$ .

$6 x + lim_{h \to 0} 3 h$

Step 5.3

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

$6 x + 3 lim_{h \to 0} h$

Step 6

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

$6 x + 3 \cdot 0$

Step 7

Simplify the answer.

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Step 7.1

Multiply $3$ by $0$ .

$6 x + 0$

Step 7.2

Add $6 x$ and $0$ .

$6 x$

Step 8