Calculus Examples

y = x^{2} + x - 1

$y = x^{2} + x - 1$ ;

x = 2

$x = 2$

Step 1

Find the corresponding

y

$y$ -value to

x = 2

$x = 2$ .

Tap for more steps...

Step 1.1

Substitute

2

$2$ in for

x

$x$ .

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

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

Step 1.2

Solve for

y

$y$ .

Tap for more steps...

Step 1.2.1

Remove parentheses.

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

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

Step 1.2.2

Remove parentheses.

y = 2^{2} + 2 - 1

$y = 2^{2} + 2 - 1$

Step 1.2.3

Remove parentheses.

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

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

Step 1.2.4

Simplify

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

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

Tap for more steps...

Step 1.2.4.1

Raise

2

$2$ to the power of

2

$2$ .

y = 4 + 2 - 1

$y = 4 + 2 - 1$

Step 1.2.4.2

Add

4

$4$ and

2

$2$ .

y = 6 - 1

$y = 6 - 1$

Step 1.2.4.3

Subtract

1

$1$ from

6

$6$ .

y = 5

$y = 5$

y = 5

$y = 5$

y = 5

$y = 5$

y = 5

$y = 5$

Step 2

Find the first derivative and evaluate at

x = 2

$x = 2$ and

y = 5

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

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of

x^{2} + x - 1

$x^{2} + x - 1$ with respect to

x

$x$ is

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

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

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

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

Step 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 = 2

$n = 2$ .

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

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

Step 2.3

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

$n = 1$ .

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

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

Step 2.4

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

2 x + 1 + 0

$2 x + 1 + 0$

Step 2.5

Add

2 x + 1

$2 x + 1$ and

0

$0$ .

2 x + 1

$2 x + 1$

Step 2.6

Evaluate the derivative at

x = 2

$x = 2$ .

2 (2) + 1

$2 (2) + 1$

Step 2.7

Simplify.

Tap for more steps...

Step 2.7.1

Multiply

2

$2$ by

2

$2$ .

4 + 1

$4 + 1$

Step 2.7.2

Add

4

$4$ and

1

$1$ .

5

$5$

5

$5$

5

$5$

Step 3

Plug the slope and point values into the point-slope formula and solve for

y

$y$ .

Tap for more steps...

Step 3.1

Use the slope

5

$5$ and a given point

(2, 5)

$(2, 5)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

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

$y - (5) = 5 \cdot (x - (2))$

Step 3.2

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

$y - 5 = 5 \cdot (x - 2)$

Step 3.3

Solve for

$y$ .

Tap for more steps...

Step 3.3.1

Simplify

$5 \cdot (x - 2)$ .

Tap for more steps...

Step 3.3.1.1

Rewrite.

$y - 5 = 0 + 0 + 5 \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 5 = 5 \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - 5 = 5 x + 5 \cdot - 2$

Step 3.3.1.4

Multiply

$5$ by

$- 2$ .

$y - 5 = 5 x - 10$

Step 3.3.2

Move all terms not containing

$y$ to the right side of the equation.

Tap for more steps...

Step 3.3.2.1

Add

$5$ to both sides of the equation.

$y = 5 x - 10 + 5$

Step 3.3.2.2

Add

$- 10$ and

$5$ .

$y = 5 x - 5$

Step 4