Calculus Examples

$y = 3 x^{2} + 3 x$ , $(1, 6)$

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

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

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.

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

Simplify the result.

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Simplify each term.

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Remove parentheses.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ by adding the exponents.

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Combine $x$ and $x$

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Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x + h) = 3 (x^{1 + 1} + x h + h x + h h) + 3 (x + h)$

Add $1$ and $1$ .

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

Multiply $h$ by $h$ by adding the exponents.

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Combine $h$ and $h$

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Raise $h$ to the power of $1$ .

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

Raise $h$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x + h) = 3 (x^{2} + x h + h x + h^{1 + 1}) + 3 (x + h)$

Add $1$ and $1$ .

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

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

Multiply $2$ by $3$ .

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

Remove parentheses around $h x$ .

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

Apply the distributive property.

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

Remove unnecessary parentheses.

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

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

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

Find the components of the definition.

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

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

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

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

Plug in the components.

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

Simplify.

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Simplify the numerator.

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Apply the distributive property.

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

Multiply $3$ by $- 1$ .

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

Multiply $3$ by $- 1$ .

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

Remove unnecessary parentheses.

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

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

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

Add $6 h x$ and $0$ .

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

Subtract $3 x$ from $3 x$ .

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

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

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

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

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

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

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

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

Factor $3 h$ out of $3 h$ .

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

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

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

Factor $3 h$ out of $3 h (2 x + h) + 3 h \cdot 1$ .

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

Simplify terms.

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Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

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

$f' (x) = lim_{h \to 0} 3 (2 x + h + 1)$

Apply the distributive property.

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

Simplify.

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Multiply $2$ by $3$ .

$f' (x) = lim_{h \to 0} 6 x + 3 h + 3 \cdot 1$

Multiply $3$ by $1$ .

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

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 $0$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

$6 \cdot x + 3 \cdot 0 + 3$

Simplify the answer.

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Simplify each term.

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Multiply $6$ by $x$ .

$6 x + 3 \cdot 0 + 3$

Multiply $3$ by $0$ .

$6 x + 0 + 3$

Add $6 x$ and $0$ .

$6 x + 3$

Simplify the right side.

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Multiply $6$ by $1$ .

$m = 6 + 3$

Add $6$ and $3$ .

$m = 9$

The slope is $m = 9$ and the point is $(1, 6)$ .

$m = 9, (1, 6)$

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.

$y = (9) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (9) \cdot (1) + b$

Substitute the value of $y$ into the equation.

$(6) = (9) \cdot (1) + b$

Find the value of $b$ .

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Rewrite the equation as $(9) \cdot (1) + b = 6$ .

$(9) \cdot (1) + b = 6$

Simplify each term.

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Multiply $9$ by $1$ .

$(9) (1) + b = 6$

Multiply $9$ by $1$ .

$9 + b = 6$

Move all terms not containing $b$ to the right side of the equation.

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Subtract $9$ from both sides of the equation.

$b = - 9 + 6$

Add $- 9$ and $6$ .

$b = - 3$

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.

$y = 9 x - 3$

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