Calculus Examples

$f (x) = 2 x^{3} + 3 x - 3$ at $x = 1$

Step 1

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

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Step 1.1

Substitute $1$ in for $x$ .

$y = 2 {(1)}^{3} + 3 (1) - 3$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2 \cdot 1^{3} + 3 (1) - 3$

Step 1.2.2

Remove parentheses.

$y = 2 {(1)}^{3} + 3 (1) - 3$

Step 1.2.3

Simplify $2 {(1)}^{3} + 3 (1) - 3$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

One to any power is one.

$y = 2 \cdot 1 + 3 (1) - 3$

Step 1.2.3.1.2

Multiply $2$ by $1$ .

$y = 2 + 3 (1) - 3$

Step 1.2.3.1.3

Multiply $3$ by $1$ .

$y = 2 + 3 - 3$

Step 1.2.3.2

Simplify by adding and subtracting.

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Step 1.2.3.2.1

Add $2$ and $3$ .

$y = 5 - 3$

Step 1.2.3.2.2

Subtract $3$ from $5$ .

$y = 2$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 2$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $1$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 2.4.2

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

$6 x^{2} + 3$

Step 2.5

Evaluate the derivative at $x = 1$ .

$6 {(1)}^{2} + 3$

Step 2.6

Simplify.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

One to any power is one.

$6 \cdot 1 + 3$

Step 2.6.1.2

Multiply $6$ by $1$ .

$6 + 3$

Step 2.6.2

Add $6$ and $3$ .

$9$

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 $(1, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 9 \cdot (x - (1))$

Step 3.2

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

$y - 2 = 9 \cdot (x - 1)$

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.

$y - 2 = 0 + 0 + 9 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 2 = 9 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y - 2 = 9 x + 9 \cdot - 1$

Step 3.3.1.4

Multiply $9$ by $- 1$ .

$y - 2 = 9 x - 9$

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

Add $2$ to both sides of the equation.

$y = 9 x - 9 + 2$

Step 3.3.2.2

Add $- 9$ and $2$ .

$y = 9 x - 7$

Step 4