Calculus Examples

$y = x^{3}$ , $(2, 8)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 8$ to find the slope of the tangent line.

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Step 1.1

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 1.2

Evaluate the derivative at $x = 2$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

$3 \cdot 4$

Step 1.3.2

Multiply $3$ by $4$ .

$12$

Step 2

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

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Step 2.1

Use the slope $12$ and a given point $(2, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = 12 \cdot (x - (2))$

Step 2.2

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

$y - 8 = 12 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $12 \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 8 = 0 + 0 + 12 \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 8 = 12 \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 8 = 12 x + 12 \cdot - 2$

Step 2.3.1.4

Multiply $12$ by $- 2$ .

$y - 8 = 12 x - 24$

Step 2.3.2

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

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Step 2.3.2.1

Add $8$ to both sides of the equation.

$y = 12 x - 24 + 8$

Step 2.3.2.2

Add $- 24$ and $8$ .

$y = 12 x - 16$

Step 3