Calculus Examples

$f (x) = x^{3} + x + 3$ ;, $(- 2, - 7)$

Step 1

Find the first derivative and evaluate at $x = - 2$ and $y = - 7$ to find the slope of the tangent line.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

$3 x^{2} + 1 + 0$

Step 1.5

Add $3 x^{2} + 1$ and $0$ .

$3 x^{2} + 1$

Step 1.6

Evaluate the derivative at $x = - 2$ .

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

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

$3 \cdot 4 + 1$

Step 1.7.1.2

Multiply $3$ by $4$ .

$12 + 1$

Step 1.7.2

Add $12$ and $1$ .

$13$

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 $13$ and a given point $(- 2, - 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 7) = 13 \cdot (x - (- 2))$

Step 2.2

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

$y + 7 = 13 \cdot (x + 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $13 \cdot (x + 2)$ .

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Step 2.3.1.1

Rewrite.

$y + 7 = 0 + 0 + 13 \cdot (x + 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 7 = 13 \cdot (x + 2)$

Step 2.3.1.3

Apply the distributive property.

$y + 7 = 13 x + 13 \cdot 2$

Step 2.3.1.4

Multiply $13$ by $2$ .

$y + 7 = 13 x + 26$

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

Subtract $7$ from both sides of the equation.

$y = 13 x + 26 - 7$

Step 2.3.2.2

Subtract $7$ from $26$ .

$y = 13 x + 19$

Step 3