Calculus Examples

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

Step 1

Find the corresponding $y$ -value to $x = 3$ .

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Step 1.1

Substitute $3$ in for $x$ .

$y = {(3)}^{3}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 3^{3}$

Step 1.2.2

Remove parentheses.

$y = {(3)}^{3}$

Step 1.2.3

Raise $3$ to the power of $3$ .

$y = 27$

Step 2

Find the first derivative and evaluate at $x = 3$ and $y = 27$ to find the slope of the tangent line.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2}$

Step 2.2

Evaluate the derivative at $x = 3$ .

$3 {(3)}^{2}$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 2.3.1.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 2.3.1.1.1

Raise $3$ to the power of $1$ .

$3^{1} {(3)}^{2}$

Step 2.3.1.1.2

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

$3^{1 + 2}$

Step 2.3.1.2

Add $1$ and $2$ .

$3^{3}$

Step 2.3.2

Raise $3$ to the power of $3$ .

$27$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $27$ and a given point $(3, 27)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (27) = 27 \cdot (x - (3))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 27 = 27 \cdot (x - 3)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $27 \cdot (x - 3)$ .

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Step 3.3.1.1

Rewrite.

$y - 27 = 0 + 0 + 27 \cdot (x - 3)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 27 = 27 \cdot (x - 3)$

Step 3.3.1.3

Apply the distributive property.

$y - 27 = 27 x + 27 \cdot - 3$

Step 3.3.1.4

Multiply $27$ by $- 3$ .

$y - 27 = 27 x - 81$

Step 3.3.2

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

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Step 3.3.2.1

Add $27$ to both sides of the equation.

$y = 27 x - 81 + 27$

Step 3.3.2.2

Add $- 81$ and $27$ .

$y = 27 x - 54$

Step 4