Calculus Examples

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

Consider the limit definition of the derivative.

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

Find the components of the definition.

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Evaluate the function at $x = x + h$ .

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Replace the variable $x$ with $x + h$ in the expression.

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

Simplify the result.

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Simplify each term.

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

$f (x + h) = (x + h) (x + h) + 2 (x + h)$

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

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Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) + 2 (x + h)$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) + 2 (x + h)$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h + 2 (x + h)$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

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

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

Reorder.

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Move $2 x$ .

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

Move $x^{2}$ .

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

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

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

Find the components of the definition.

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

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

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

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

Plug in the components.

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

Simplify.

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Simplify the numerator.

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Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

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

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

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

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

Subtract $2 x$ from $2 x$ .

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

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

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

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

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

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

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

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

Factor $h$ out of $2 h$ .

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

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

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

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

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor of $h$ .

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Cancel the common factor.

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

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

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

Reorder $h$ and $2 x$ .

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

Take the limit of each term.

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

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

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

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

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

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

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

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

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

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

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

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

$2 \cdot x + 0 + 2$

Add $2 \cdot x$ and $0$ .

$2 x + 2$

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