Calculus Examples

$f (x) = x^{3} - 3 x^{2} + 2 x - 2$ at $x = 0$

Step 1

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

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Step 1.1

Substitute $0$ in for $x$ .

$y = {(0)}^{3} - 3 {(0)}^{2} + 2 (0) - 2$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 0^{3} - 3 {(0)}^{2} + 2 (0) - 2$

Step 1.2.2

Remove parentheses.

$y = 0^{3} - 3 \cdot 0^{2} + 2 (0) - 2$

Step 1.2.3

Remove parentheses.

$y = {(0)}^{3} - 3 {(0)}^{2} + 2 (0) - 2$

Step 1.2.4

Simplify ${(0)}^{3} - 3 {(0)}^{2} + 2 (0) - 2$ .

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Step 1.2.4.1

Simplify each term.

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Step 1.2.4.1.1

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

$y = 0 - 3 {(0)}^{2} + 2 (0) - 2$

Step 1.2.4.1.2

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

$y = 0 - 3 \cdot 0 + 2 (0) - 2$

Step 1.2.4.1.3

Multiply $- 3$ by $0$ .

$y = 0 + 0 + 2 (0) - 2$

Step 1.2.4.1.4

Multiply $2$ by $0$ .

$y = 0 + 0 + 0 - 2$

Step 1.2.4.2

Simplify by adding and subtracting.

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Step 1.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 - 2$

Step 1.2.4.2.2

Add $0$ and $0$ .

$y = 0 - 2$

Step 1.2.4.2.3

Subtract $2$ from $0$ .

$y = - 2$

Step 2

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

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $- 3$ .

$3 x^{2} - 6 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 2]$

Step 2.3

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

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Step 2.3.1

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

$3 x^{2} - 6 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 2.3.2

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

$3 x^{2} - 6 x + 2 \cdot 1 + \frac{d}{d x} [- 2]$

Step 2.3.3

Multiply $2$ by $1$ .

$3 x^{2} - 6 x + 2 + \frac{d}{d x} [- 2]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$3 x^{2} - 6 x + 2 + 0$

Step 2.4.2

Add $3 x^{2} - 6 x + 2$ and $0$ .

$3 x^{2} - 6 x + 2$

Step 2.5

Evaluate the derivative at $x = 0$ .

$3 {(0)}^{2} - 6 \cdot 0 + 2$

Step 2.6

Simplify.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

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

$3 \cdot 0 - 6 \cdot 0 + 2$

Step 2.6.1.2

Multiply $3$ by $0$ .

$0 - 6 \cdot 0 + 2$

Step 2.6.1.3

Multiply $- 6$ by $0$ .

$0 + 0 + 2$

Step 2.6.2

Simplify by adding numbers.

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Step 2.6.2.1

Add $0$ and $0$ .

$0 + 2$

Step 2.6.2.2

Add $0$ and $2$ .

$2$

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

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

Add $x$ and $0$ .

$y + 2 = 2 x$

Step 3.3.2

Subtract $2$ from both sides of the equation.

$y = 2 x - 2$

Step 4