Calculus Examples

$f (x) = 3 x^{3}$ , $x = - 1$

Step 1

Find the corresponding $y$ -value to $x = - 1$ .

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Step 1.1

Substitute $- 1$ in for $x$ .

$y = 3 {(- 1)}^{3}$

Step 1.2

Simplify $3 {(- 1)}^{3}$ .

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Step 1.2.1

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

$y = 3 \cdot - 1$

Step 1.2.2

Multiply $3$ by $- 1$ .

$y = - 3$

Step 2

Find the first derivative and evaluate at $x = - 1$ and $y = - 3$ to find the slope of the tangent line.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

$3 \frac{d}{d x} [x^{3}]$

Step 2.2

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

$3 (3 x^{2})$

Step 2.3

Multiply $3$ by $3$ .

$9 x^{2}$

Step 2.4

Evaluate the derivative at $x = - 1$ .

$9 {(- 1)}^{2}$

Step 2.5

Simplify.

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Step 2.5.1

Raise $- 1$ to the power of $2$ .

$9 \cdot 1$

Step 2.5.2

Multiply $9$ by $1$ .

$9$

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 $9$ and a given point $(- 1, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = 9 \cdot (x - (- 1))$

Step 3.2

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

$y + 3 = 9 \cdot (x + 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $9 \cdot (x + 1)$ .

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Step 3.3.1.1

Rewrite.

$y + 3 = 0 + 0 + 9 \cdot (x + 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y + 3 = 9 \cdot (x + 1)$

Step 3.3.1.3

Apply the distributive property.

$y + 3 = 9 x + 9 \cdot 1$

Step 3.3.1.4

Multiply $9$ by $1$ .

$y + 3 = 9 x + 9$

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

Subtract $3$ from both sides of the equation.

$y = 9 x + 9 - 3$

Step 3.3.2.2

Subtract $3$ from $9$ .

$y = 9 x + 6$

Step 4