Calculus Examples

f (x) = x^{3} - 7 x + 1

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

(1, - 5)

$(1, - 5)$

Step 1

Find the first derivative and evaluate at

x = 1

$x = 1$ and

y = - 5

$y = - 5$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of

x^{3} - 7 x + 1

$x^{3} - 7 x + 1$ with respect to

x

$x$ is

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

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

\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}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

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

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

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

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

Step 1.2

Evaluate

\frac{d}{d x} [- 7 x]

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

Tap for more steps...

Step 1.2.1

Since

- 7

$- 7$ is constant with respect to

x

$x$ , the derivative of

- 7 x

$- 7 x$ with respect to

x

$x$ is

- 7 \frac{d}{d x} [x]

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

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

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

Step 1.2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 1.2.3

Multiply

- 7

$- 7$ by

1

$1$ .

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

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

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

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

Step 1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.3.1

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

3 x^{2} - 7 + 0

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

Step 1.3.2

Add

3 x^{2} - 7

$3 x^{2} - 7$ and

0

$0$ .

3 x^{2} - 7

$3 x^{2} - 7$

3 x^{2} - 7

$3 x^{2} - 7$

Step 1.4

Evaluate the derivative at

x = 1

$x = 1$ .

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

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Simplify each term.

Tap for more steps...

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

$y$ .

Tap for more steps...

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

$y$ .

Tap for more steps...

Step 2.3.1

Simplify

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

Tap for more steps...

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.

Tap for more steps...

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