Calculus Examples

$y = x^{2} + 3 x + 34$ , $(0, 34)$

The slope of the tangent line is the derivative of the expression.

$m$ $=$ The derivative of $y = x^{2} + 3 x + 34$

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.

$f (x + h) = {(x + h)}^{2} + 3 (x + h) + 34$

Simplify the result.

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Simplify each term.

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

$f (x + h) = (x + h) (x + h) + 3 (x + h) + 34$

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

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Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) + 3 (x + h) + 34$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) + 3 (x + h) + 34$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h + 3 (x + h) + 34$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h + 3 (x + h) + 34$

Multiply $h$ by $h$ .

$f (x + h) = x^{2} + x h + h x + h^{2} + 3 (x + h) + 34$

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

$f (x + h) = x^{2} + (h x + h x) + h^{2} + 3 (x + h) + 34$

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 3 (x + h) + 34$

Apply the distributive property.

$f (x + h) = x^{2} + 2 h x + h^{2} + (3 x + 3 h) + 34$

$f (x + h) = x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$

The final answer is $x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$ .

$x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$

Find the components of the definition.

$f (x + h) = x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$

$f (x) = x^{2} + 3 x + 34$

$f (x + h) = x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$

$f (x) = x^{2} + 3 x + 34$

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34 - (x^{2} + 3 x + 34)}{h}$

Simplify.

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Simplify the numerator.

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Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34 + (- x^{2} - (3 x) - 1 \cdot 34)}{h}$

Simplify.

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Multiply $3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34 + (- x^{2} - 3 x - 1 \cdot 34)}{h}$

Multiply $- 1$ by $34$ .

$f' (x) = lim_{h \to 0} \frac{x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34 + (- x^{2} - 3 x - 34)}{h}$

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

$f' (x) = lim_{h \to 0} \frac{2 h x + h^{2} + 3 x + 3 h + 34 + 0 - 3 x - 34}{h}$

Add $2 h x$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{2 h x + h^{2} + 3 x + 3 h + 34 - 3 x - 34}{h}$

Subtract $3 x$ from $3 x$ .

$f' (x) = lim_{h \to 0} \frac{2 h x + h^{2} + 3 h + 34 + 0 - 34}{h}$

Add $2 h x$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{2 h x + h^{2} + 3 h + 34 - 34}{h}$

Subtract $34$ from $34$ .

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

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

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

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

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

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

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

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

Factor $h$ out of $3 h$ .

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

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

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

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

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

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

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

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 $0$ .

$lim_{h \to 0} 2 x + lim_{h \to 0} h + lim_{h \to 0} 3$

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

$lim_{h \to 0} 2 \cdot lim_{h \to 0} x + lim_{h \to 0} h + lim_{h \to 0} 3$

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 $0$ .

$2 \cdot lim_{h \to 0} x + lim_{h \to 0} h + lim_{h \to 0} 3$

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

$2 \cdot x + lim_{h \to 0} h + lim_{h \to 0} 3$

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

$2 \cdot x + 0 + lim_{h \to 0} 3$

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$2 \cdot x + 0 + 3$

Add $2 x$ and $0$ .

$2 x + 3$

Simplify $2 (0) + 3$ .

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Multiply $2$ by $0$ .

$m = 0 + 3$

Add $0$ and $3$ .

$m = 3$

The slope is $m = 3$ and the point is $(0, 34)$ .

$m = 3, (0, 34)$

Find the value of $b$ using the formula for the equation of a line.

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Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (3) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (3) \cdot (0) + b$

Substitute the value of $y$ into the equation.

$34 = (3) \cdot (0) + b$

Find the value of $b$ .

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Rewrite the equation as $(3) \cdot (0) + b = 34$ .

$(3) \cdot (0) + b = 34$

Simplify the left side.

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Multiply $3$ by $0$ .

$0 + b = 34$

Add $0$ and $b$ .

$b = 34$

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = 3 x + 34$

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