Calculus Examples

$f (x) = 2 x^{2} - 1$ ; $x = 0$

Step 1

Find the corresponding $y$ -value to $x = 0$ .

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Step 1.1

Substitute $0$ in for $x$ .

$y = 2 {(0)}^{2} - 1$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2 \cdot 0^{2} - 1$

Step 1.2.2

Remove parentheses.

$y = 2 {(0)}^{2} - 1$

Step 1.2.3

Simplify $2 {(0)}^{2} - 1$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

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

$y = 2 \cdot 0 - 1$

Step 1.2.3.1.2

Multiply $2$ by $0$ .

$y = 0 - 1$

Step 1.2.3.2

Subtract $1$ from $0$ .

$y = - 1$

Step 2

Find the first derivative and evaluate at $x = 0$ and $y = - 1$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 1]$

Step 2.2

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

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Step 2.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 (2 x) + \frac{d}{d x} [- 1]$

Step 2.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 1]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$4 x + 0$

Step 2.3.2

Add $4 x$ and $0$ .

$4 x$

Step 2.4

Evaluate the derivative at $x = 0$ .

$4 (0)$

Step 2.5

Multiply $4$ by $0$ .

$0$

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

$y - (- 1) = 0 \cdot (x - (0))$

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $0 \cdot (x + 0)$ .

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Step 3.3.1.1

Add $x$ and $0$ .

$y + 1 = 0 \cdot x$

Step 3.3.1.2

Multiply $0$ by $x$ .

$y + 1 = 0$

Step 3.3.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 4