Calculus Examples

$f (x) = x^{3} - 7 x + 1$ ; $(1, - 5)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = - 5$ to find the slope of the tangent line.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{3} - 7 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [1]$

Step 1.1.2

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

$3 x^{2} + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [1]$

Step 1.2

Evaluate $\frac{d}{d x} [- 7 x]$ .

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Step 1.2.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$3 x^{2} - 7 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.2.2

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

$3 x^{2} - 7 \cdot 1 + \frac{d}{d x} [1]$

Step 1.2.3

Multiply $- 7$ by $1$ .

$3 x^{2} - 7 + \frac{d}{d x} [1]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

$3 x^{2} - 7 + 0$

Step 1.3.2

Add $3 x^{2} - 7$ and $0$ .

$3 x^{2} - 7$

Step 1.4

Evaluate the derivative at $x = 1$ .

$3 {(1)}^{2} - 7$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$3 \cdot 1 - 7$

Step 1.5.1.2

Multiply $3$ by $1$ .

$3 - 7$

Step 1.5.2

Subtract $7$ from $3$ .

$- 4$

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 $- 4$ and a given point $(1, - 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 5) = - 4 \cdot (x - (1))$

Step 2.2

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

$y + 5 = - 4 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 4 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y + 5 = 0 + 0 - 4 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 5 = - 4 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y + 5 = - 4 x - 4 \cdot - 1$

Step 2.3.1.4

Multiply $- 4$ by $- 1$ .

$y + 5 = - 4 x + 4$

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 $5$ from both sides of the equation.

$y = - 4 x + 4 - 5$

Step 2.3.2.2

Subtract $5$ from $4$ .

$y = - 4 x - 1$

Step 3