Calculus Examples

$y = 5 x^{3} - 3 x$ , $(1, 2)$

Step 1

Write $y = 5 x^{3} - 3 x$ as a function.

$f (x) = 5 x^{3} - 3 x$

Step 2

Check if the given point is on the graph of the given function.

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Step 2.1

Evaluate $f (x) = 5 x^{3} - 3 x$ at $x = 1$ .

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Step 2.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 5 {(1)}^{3} - 3 \cdot 1$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

One to any power is one.

$f (1) = 5 \cdot 1 - 3 \cdot 1$

Step 2.1.2.1.2

Multiply $5$ by $1$ .

$f (1) = 5 - 3 \cdot 1$

Step 2.1.2.1.3

Multiply $- 3$ by $1$ .

$f (1) = 5 - 3$

Step 2.1.2.2

Subtract $3$ from $5$ .

$f (1) = 2$

Step 2.1.2.3

The final answer is $2$ .

$2$

Step 2.2

Since $2 = 2$ , the point is on the graph.

The point is on the graph

Step 3

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

$m$ $=$ The derivative of $f (x) = 5 x^{3} - 3 x$

Step 4

Consider the limit definition of the derivative.

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

Step 5

Find the components of the definition.

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Step 5.1

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

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Step 5.1.1

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

$f (x + h) = 5 {(x + h)}^{3} - 3 (x + h)$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Use the Binomial Theorem.

$f (x + h) = 5 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) - 3 (x + h)$

Step 5.1.2.1.2

Apply the distributive property.

$f (x + h) = 5 x^{3} + 5 (3 x^{2} h) + 5 (3 x h^{2}) + 5 h^{3} - 3 (x + h)$

Step 5.1.2.1.3

Simplify.

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Step 5.1.2.1.3.1

Multiply $3$ by $5$ .

$f (x + h) = 5 x^{3} + 15 (x^{2} h) + 5 (3 x h^{2}) + 5 h^{3} - 3 (x + h)$

Step 5.1.2.1.3.2

Multiply $3$ by $5$ .

$f (x + h) = 5 x^{3} + 15 (x^{2} h) + 15 (x h^{2}) + 5 h^{3} - 3 (x + h)$

Step 5.1.2.1.4

Remove parentheses.

$f (x + h) = 5 x^{3} + 15 x^{2} h + 15 x h^{2} + 5 h^{3} - 3 (x + h)$

Step 5.1.2.1.5

Apply the distributive property.

$f (x + h) = 5 x^{3} + 15 x^{2} h + 15 x h^{2} + 5 h^{3} - 3 x - 3 h$

Step 5.1.2.2

The final answer is $5 x^{3} + 15 x^{2} h + 15 x h^{2} + 5 h^{3} - 3 x - 3 h$ .

$5 x^{3} + 15 x^{2} h + 15 x h^{2} + 5 h^{3} - 3 x - 3 h$

Step 5.2

Reorder.

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Step 5.2.1

Move $x^{2}$ .

$5 x^{3} + 15 h x^{2} + 15 x h^{2} + 5 h^{3} - 3 x - 3 h$

Step 5.2.2

Move $x$ .

$5 x^{3} + 15 h x^{2} + 15 h^{2} x + 5 h^{3} - 3 x - 3 h$

Step 5.2.3

Move $- 3 x$ .

$5 x^{3} + 15 h x^{2} + 15 h^{2} x + 5 h^{3} - 3 h - 3 x$

Step 5.2.4

Move $5 x^{3}$ .

$15 h x^{2} + 15 h^{2} x + 5 h^{3} + 5 x^{3} - 3 h - 3 x$

Step 5.2.5

Move $15 h x^{2}$ .

$15 h^{2} x + 5 h^{3} + 15 h x^{2} + 5 x^{3} - 3 h - 3 x$

Step 5.2.6

Reorder $15 h^{2} x$ and $5 h^{3}$ .

$5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} - 3 h - 3 x$

Step 5.3

Find the components of the definition.

$f (x + h) = 5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} - 3 h - 3 x$

$f (x) = 5 x^{3} - 3 x$

$f (x + h) = 5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} - 3 h - 3 x$

$f (x) = 5 x^{3} - 3 x$

Step 6

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} - 3 h - 3 x - (5 x^{3} - 3 x)}{h}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} - 3 h - 3 x - (5 x^{3}) - (- 3 x)}{h}$

Step 7.1.2

Multiply $5$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} - 3 h - 3 x - 5 x^{3} - (- 3 x)}{h}$

Step 7.1.3

Multiply $- 3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} + 5 x^{3} - 3 h - 3 x - 5 x^{3} + 3 x}{h}$

Step 7.1.4

Subtract $5 x^{3}$ from $5 x^{3}$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} - 3 h - 3 x + 0 + 3 x}{h}$

Step 7.1.5

Add $5 h^{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} - 3 h - 3 x + 3 x}{h}$

Step 7.1.6

Add $- 3 x$ and $3 x$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} - 3 h + 0}{h}$

Step 7.1.7

Add $5 h^{3} + 15 h^{2} x + 15 h x^{2} - 3 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{5 h^{3} + 15 h^{2} x + 15 h x^{2} - 3 h}{h}$

Step 7.1.8

Factor $h$ out of $5 h^{3} + 15 h^{2} x + 15 h x^{2} - 3 h$ .

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Step 7.1.8.1

Factor $h$ out of $5 h^{3}$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + 15 h^{2} x + 15 h x^{2} - 3 h}{h}$

Step 7.1.8.2

Factor $h$ out of $15 h^{2} x$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + h (15 h x) + 15 h x^{2} - 3 h}{h}$

Step 7.1.8.3

Factor $h$ out of $15 h x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + h (15 h x) + h (15 x^{2}) - 3 h}{h}$

Step 7.1.8.4

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

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2}) + h (15 h x) + h (15 x^{2}) + h \cdot - 3}{h}$

Step 7.1.8.5

Factor $h$ out of $h (5 h^{2}) + h (15 h x)$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x) + h (15 x^{2}) + h \cdot - 3}{h}$

Step 7.1.8.6

Factor $h$ out of $h (5 h^{2} + 15 h x) + h (15 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x + 15 x^{2}) + h \cdot - 3}{h}$

Step 7.1.8.7

Factor $h$ out of $h (5 h^{2} + 15 h x + 15 x^{2}) + h \cdot - 3$ .

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x + 15 x^{2} - 3)}{h}$

Step 7.2

Reduce the expression by cancelling the common factors.

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Step 7.2.1

Cancel the common factor of $h$ .

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Step 7.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (5 h^{2} + 15 h x + 15 x^{2} - 3)}{h}$

Step 7.2.1.2

Divide $5 h^{2} + 15 h x + 15 x^{2} - 3$ by $1$ .

$f' (x) = lim_{h \to 0} 5 h^{2} + 15 h x + 15 x^{2} - 3$

Step 7.2.2

Simplify the expression.

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Step 7.2.2.1

Move $h$ .

$f' (x) = lim_{h \to 0} 5 h^{2} + 15 x h + 15 x^{2} - 3$

Step 7.2.2.2

Move $5 h^{2}$ .

$f' (x) = lim_{h \to 0} 15 x h + 15 x^{2} + 5 h^{2} - 3$

Step 7.2.2.3

Reorder $15 x h$ and $15 x^{2}$ .

$f' (x) = lim_{h \to 0} 15 x^{2} + 15 x h + 5 h^{2} - 3$

Step 8

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

$lim_{h \to 0} 15 x^{2} + lim_{h \to 0} 15 x h + lim_{h \to 0} 5 h^{2} - lim_{h \to 0} 3$

Step 9

Evaluate the limit of $15 x^{2}$ which is constant as $h$ approaches $0$ .

$15 x^{2} + lim_{h \to 0} 15 x h + lim_{h \to 0} 5 h^{2} - lim_{h \to 0} 3$

Step 10

Move the term $15 x$ outside of the limit because it is constant with respect to $h$ .

$15 x^{2} + 15 x lim_{h \to 0} h + lim_{h \to 0} 5 h^{2} - lim_{h \to 0} 3$

Step 11

Move the term $5$ outside of the limit because it is constant with respect to $h$ .

$15 x^{2} + 15 x lim_{h \to 0} h + 5 lim_{h \to 0} h^{2} - lim_{h \to 0} 3$

Step 12

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

$15 x^{2} + 15 x lim_{h \to 0} h + 5 {(lim_{h \to 0} h)}^{2} - lim_{h \to 0} 3$

Step 13

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

$15 x^{2} + 15 x lim_{h \to 0} h + 5 {(lim_{h \to 0} h)}^{2} - 1 \cdot 3$

Step 14

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

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Step 14.1

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

$15 x^{2} + 15 x \cdot 0 + 5 {(lim_{h \to 0} h)}^{2} - 1 \cdot 3$

Step 14.2

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

$15 x^{2} + 15 x \cdot 0 + 5 \cdot 0^{2} - 1 \cdot 3$

Step 15

Simplify the answer.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $15 x \cdot 0$ .

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Step 15.1.1.1

Multiply $0$ by $15$ .

$15 x^{2} + 0 x + 5 \cdot 0^{2} - 1 \cdot 3$

Step 15.1.1.2

Multiply $0$ by $x$ .

$15 x^{2} + 0 + 5 \cdot 0^{2} - 1 \cdot 3$

Step 15.1.2

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

$15 x^{2} + 0 + 5 \cdot 0 - 1 \cdot 3$

Step 15.1.3

Multiply $5$ by $0$ .

$15 x^{2} + 0 + 0 - 1 \cdot 3$

Step 15.1.4

Multiply $- 1$ by $3$ .

$15 x^{2} + 0 + 0 - 3$

Step 15.2

Combine the opposite terms in $15 x^{2} + 0 + 0 - 3$ .

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Step 15.2.1

Add $15 x^{2}$ and $0$ .

$15 x^{2} + 0 - 3$

Step 15.2.2

Add $15 x^{2}$ and $0$ .

$15 x^{2} - 3$

Step 16

Find the slope $m$ . In this case $m = 12$ .

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Step 16.1

Remove parentheses.

$m = 15 \cdot 1^{2} - 3$

Step 16.2

Simplify $15 \cdot 1^{2} - 3$ .

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

One to any power is one.

$m = 15 \cdot 1 - 3$

Step 16.2.1.2

Multiply $15$ by $1$ .

$m = 15 - 3$

Step 16.2.2

Subtract $3$ from $15$ .

$m = 12$

Step 17

The slope is $m = 12$ and the point is $(1, 2)$ .

$m = 12, (1, 2)$

Step 18

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

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Step 18.1

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

$y = m x + b$

Step 18.2

Substitute the value of $m$ into the equation.

$y = (12) \cdot x + b$

Step 18.3

Substitute the value of $x$ into the equation.

$y = (12) \cdot (1) + b$

Step 18.4

Substitute the value of $y$ into the equation.

$2 = (12) \cdot (1) + b$

Step 18.5

Find the value of $b$ .

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Step 18.5.1

Rewrite the equation as $(12) \cdot (1) + b = 2$ .

$(12) \cdot (1) + b = 2$

Step 18.5.2

Multiply $12$ by $1$ .

$12 + b = 2$

Step 18.5.3

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

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Step 18.5.3.1

Subtract $12$ from both sides of the equation.

$b = 2 - 12$

Step 18.5.3.2

Subtract $12$ from $2$ .

$b = - 10$

Step 19

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.

$y = 12 x - 10$

Step 20