Calculus Examples

$y = 3 x^{3} + x + 3$ , $(1, 7)$

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

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

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.

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

Simplify the result.

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Remove parentheses.

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

Simplify each term.

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Use the Binomial Theorem.

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

Apply the distributive property.

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

Simplify.

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Multiply $3$ by $3$ .

$f (x + h) = 3 x^{3} + 9 (x^{2} h) + 3 (3 x h^{2}) + 3 h^{3} + x + h + 3$

Multiply $3$ by $3$ .

$f (x + h) = 3 x^{3} + 9 (x^{2} h) + 9 (x h^{2}) + 3 h^{3} + x + h + 3$

Remove parentheses.

$f (x + h) = 3 x^{3} + 9 x^{2} h + 9 x h^{2} + 3 h^{3} + x + h + 3$

The final answer is $3 x^{3} + 9 x^{2} h + 9 x h^{2} + 3 h^{3} + x + h + 3$ .

$3 x^{3} + 9 x^{2} h + 9 x h^{2} + 3 h^{3} + x + h + 3$

Reorder.

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Move $x^{2}$ .

$3 x^{3} + 9 h x^{2} + 9 x h^{2} + 3 h^{3} + x + h + 3$

Move $x$ .

$3 x^{3} + 9 h x^{2} + 9 h^{2} x + 3 h^{3} + x + h + 3$

Move $x$ .

$3 x^{3} + 9 h x^{2} + 9 h^{2} x + 3 h^{3} + h + x + 3$

Move $3 x^{3}$ .

$9 h x^{2} + 9 h^{2} x + 3 h^{3} + 3 x^{3} + h + x + 3$

Move $9 h x^{2}$ .

$9 h^{2} x + 3 h^{3} + 9 h x^{2} + 3 x^{3} + h + x + 3$

Reorder $9 h^{2} x$ and $3 h^{3}$ .

$3 h^{3} + 9 h^{2} x + 9 h x^{2} + 3 x^{3} + h + x + 3$

Find the components of the definition.

$f (x + h) = 3 h^{3} + 9 h^{2} x + 9 h x^{2} + 3 x^{3} + h + x + 3$

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

$f (x + h) = 3 h^{3} + 9 h^{2} x + 9 h x^{2} + 3 x^{3} + h + x + 3$

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

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + 3 x^{3} + h + x + 3 - (3 x^{3} + x + 3)}{h}$

Simplify.

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Simplify the numerator.

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Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + 3 x^{3} + h + x + 3 - (3 x^{3}) - x - 1 \cdot 3}{h}$

Simplify.

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Multiply $3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + 3 x^{3} + h + x + 3 - 3 x^{3} - x - 1 \cdot 3}{h}$

Multiply $- 1$ by $3$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + 3 x^{3} + h + x + 3 - 3 x^{3} - x - 3}{h}$

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

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + h + x + 3 + 0 - x - 3}{h}$

Add $3 h^{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + h + x + 3 - x - 3}{h}$

Subtract $x$ from $x$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + h + 0 + 3 - 3}{h}$

Add $3 h^{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + h + 3 - 3}{h}$

Subtract $3$ from $3$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + h + 0}{h}$

Add $3 h^{3} + 9 h^{2} x + 9 h x^{2} + h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{3} + 9 h^{2} x + 9 h x^{2} + h}{h}$

Factor $h$ out of $3 h^{3} + 9 h^{2} x + 9 h x^{2} + h$ .

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

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2}) + 9 h^{2} x + 9 h x^{2} + h}{h}$

Factor $h$ out of $9 h^{2} x$ .

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2}) + h (9 h x) + 9 h x^{2} + h}{h}$

Factor $h$ out of $9 h x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2}) + h (9 h x) + h (9 x^{2}) + h}{h}$

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

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2}) + h (9 h x) + h (9 x^{2}) + h}{h}$

Factor $h$ out of $h^{1}$ .

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2}) + h (9 h x) + h (9 x^{2}) + h \cdot 1}{h}$

Factor $h$ out of $h (3 h^{2}) + h (9 h x)$ .

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2} + 9 h x) + h (9 x^{2}) + h \cdot 1}{h}$

Factor $h$ out of $h (3 h^{2} + 9 h x) + h (9 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2} + 9 h x + 9 x^{2}) + h \cdot 1}{h}$

Factor $h$ out of $h (3 h^{2} + 9 h x + 9 x^{2}) + h \cdot 1$ .

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2} + 9 h x + 9 x^{2} + 1)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h (3 h^{2} + 9 h x + 9 x^{2} + 1)}{h}$

Divide $3 h^{2} + 9 h x + 9 x^{2} + 1$ by $1$ .

$f' (x) = lim_{h \to 0} 3 h^{2} + 9 h x + 9 x^{2} + 1$

Simplify the expression.

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Move $h$ .

$f' (x) = lim_{h \to 0} 3 h^{2} + 9 x h + 9 x^{2} + 1$

Move $3 h^{2}$ .

$f' (x) = lim_{h \to 0} 9 x h + 9 x^{2} + 3 h^{2} + 1$

Reorder $9 x h$ and $9 x^{2}$ .

$f' (x) = lim_{h \to 0} 9 x^{2} + 9 x h + 3 h^{2} + 1$

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 $0$ .

$lim_{h \to 0} 9 x^{2} + lim_{h \to 0} 9 x h + lim_{h \to 0} 3 h^{2} + lim_{h \to 0} 1$

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

$lim_{h \to 0} 9 \cdot lim_{h \to 0} x^{2} + lim_{h \to 0} 9 x h + lim_{h \to 0} 3 h^{2} + lim_{h \to 0} 1$

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$lim_{h \to 0} 9 \cdot {(lim_{h \to 0} x)}^{2} + lim_{h \to 0} 9 x h + lim_{h \to 0} 3 h^{2} + lim_{h \to 0} 1$

Move the term $9 x$ outside of the limit because it is constant with respect to $h$ .

$lim_{h \to 0} 9 \cdot {(lim_{h \to 0} x)}^{2} + 9 x lim_{h \to 0} h + lim_{h \to 0} 3 h^{2} + lim_{h \to 0} 1$

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

$lim_{h \to 0} 9 \cdot {(lim_{h \to 0} x)}^{2} + 9 x lim_{h \to 0} h + 3 lim_{h \to 0} h^{2} + lim_{h \to 0} 1$

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$lim_{h \to 0} 9 \cdot {(lim_{h \to 0} x)}^{2} + 9 x lim_{h \to 0} h + 3 {(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 1$

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

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Evaluate the limit of $9$ which is constant as $h$ approaches $0$ .

$9 \cdot {(lim_{h \to 0} x)}^{2} + 9 x lim_{h \to 0} h + 3 {(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 1$

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

$9 \cdot x^{2} + 9 x lim_{h \to 0} h + 3 {(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 1$

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

$9 \cdot x^{2} + 9 x \cdot 0 + 3 {(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 1$

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

$9 \cdot x^{2} + 9 x \cdot 0 + 3 \cdot 0^{2} + lim_{h \to 0} 1$

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$9 \cdot x^{2} + 9 x \cdot 0 + 3 \cdot 0^{2} + 1$

Simplify the answer.

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Simplify each term.

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Multiply $9 x \cdot 0$ .

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Multiply $0$ by $9$ .

$9 x^{2} + 0 x + 3 \cdot 0^{2} + 1$

Multiply $0$ by $x$ .

$9 x^{2} + 0 + 3 \cdot 0^{2} + 1$

Raising $0$ to any positive power yields $0$ .

$9 x^{2} + 0 + 3 \cdot 0 + 1$

Multiply $3$ by $0$ .

$9 x^{2} + 0 + 0 + 1$

Combine the opposite terms in $9 x^{2} + 0 + 0 + 1$ .

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Add $9 x^{2}$ and $0$ .

$9 x^{2} + 0 + 1$

Add $9 x^{2}$ and $0$ .

$9 x^{2} + 1$

Simplify $9 {(1)}^{2} + 1$ .

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Simplify each term.

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One to any power is one.

$m = 9 \cdot 1 + 1$

Multiply $9$ by $1$ .

$m = 9 + 1$

Add $9$ and $1$ .

$m = 10$

The slope is $m = 10$ and the point is $(1, 7)$ .

$m = 10, (1, 7)$

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.

$y = (10) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (10) \cdot (1) + b$

Substitute the value of $y$ into the equation.

$7 = (10) \cdot (1) + b$

Find the value of $b$ .

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Rewrite the equation as $(10) \cdot (1) + b = 7$ .

$(10) \cdot (1) + b = 7$

Multiply $10$ by $1$ .

$10 + b = 7$

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

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Subtract $10$ from both sides of the equation.

$b = 7 - 10$

Subtract $10$ from $7$ .

$b = - 3$

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.

$y = 10 x - 3$

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