Calculus Examples

$y = 2 x^{3} - 8 x$ , $(2, 0)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{3} - 8 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 8 x]$ .

$\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 8 x]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x]$

Step 1.2.2

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

$2 (3 x^{2}) + \frac{d}{d x} [- 8 x]$

Step 1.2.3

Multiply $3$ by $2$ .

$6 x^{2} + \frac{d}{d x} [- 8 x]$

Step 1.3

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$6 x^{2} - 8 \frac{d}{d x} [x]$

Step 1.3.2

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

$6 x^{2} - 8 \cdot 1$

Step 1.3.3

Multiply $- 8$ by $1$ .

$6 x^{2} - 8$

Step 1.4

Evaluate the derivative at $x = 2$ .

$6 {(2)}^{2} - 8$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

$6 \cdot 4 - 8$

Step 1.5.1.2

Multiply $6$ by $4$ .

$24 - 8$

Step 1.5.2

Subtract $8$ from $24$ .

$16$

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 $16$ and a given point $(2, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 16 \cdot (x - (2))$

Step 2.2

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

$y + 0 = 16 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = 16 \cdot (x - 2)$

Step 2.3.2

Simplify $16 \cdot (x - 2)$ .

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Step 2.3.2.1

Apply the distributive property.

$y = 16 x + 16 \cdot - 2$

Step 2.3.2.2

Multiply $16$ by $- 2$ .

$y = 16 x - 32$

Step 3