# Calculus Examples

,

Step 1

Write as a function.

Step 2

Step 2.1

Evaluate at .

Step 2.1.1

Replace the variable with in the expression.

Step 2.1.2

Simplify the result.

Step 2.1.2.1

Simplify each term.

Step 2.1.2.1.1

One to any power is one.

Step 2.1.2.1.2

Multiply by .

Step 2.1.2.1.3

Multiply by .

Step 2.1.2.2

Add and .

Step 2.1.2.3

The final answer is .

Step 2.2

Since , the point is on the graph.

The point is on the graph

The point is on the graph

Step 3

The slope of the tangent line is the derivative of the expression.

The derivative of

Step 4

Consider the limit definition of the derivative.

Step 5

Step 5.1

Evaluate the function at .

Step 5.1.1

Replace the variable with in the expression.

Step 5.1.2

Simplify the result.

Step 5.1.2.1

Simplify each term.

Step 5.1.2.1.1

Rewrite as .

Step 5.1.2.1.2

Expand using the FOIL Method.

Step 5.1.2.1.2.1

Apply the distributive property.

Step 5.1.2.1.2.2

Apply the distributive property.

Step 5.1.2.1.2.3

Apply the distributive property.

Step 5.1.2.1.3

Simplify and combine like terms.

Step 5.1.2.1.3.1

Simplify each term.

Step 5.1.2.1.3.1.1

Multiply by .

Step 5.1.2.1.3.1.2

Multiply by .

Step 5.1.2.1.3.2

Add and .

Step 5.1.2.1.3.2.1

Reorder and .

Step 5.1.2.1.3.2.2

Add and .

Step 5.1.2.1.4

Apply the distributive property.

Step 5.1.2.1.5

Multiply by .

Step 5.1.2.1.6

Apply the distributive property.

Step 5.1.2.2

The final answer is .

Step 5.2

Reorder.

Step 5.2.1

Move .

Step 5.2.2

Move .

Step 5.2.3

Reorder and .

Step 5.3

Find the components of the definition.

Step 6

Plug in the components.

Step 7

Step 7.1

Simplify the numerator.

Step 7.1.1

Apply the distributive property.

Step 7.1.2

Multiply by .

Step 7.1.3

Multiply by .

Step 7.1.4

Subtract from .

Step 7.1.5

Add and .

Step 7.1.6

Subtract from .

Step 7.1.7

Add and .

Step 7.1.8

Factor out of .

Step 7.1.8.1

Factor out of .

Step 7.1.8.2

Factor out of .

Step 7.1.8.3

Factor out of .

Step 7.1.8.4

Factor out of .

Step 7.1.8.5

Factor out of .

Step 7.2

Simplify terms.

Step 7.2.1

Cancel the common factor of .

Step 7.2.1.1

Cancel the common factor.

Step 7.2.1.2

Divide by .

Step 7.2.2

Apply the distributive property.

Step 7.3

Simplify.

Step 7.3.1

Multiply by .

Step 7.3.2

Multiply by .

Step 7.4

Reorder and .

Step 8

Step 8.1

Split the limit using the Sum of Limits Rule on the limit as approaches .

Step 8.2

Evaluate the limit of which is constant as approaches .

Step 8.3

Move the term outside of the limit because it is constant with respect to .

Step 8.4

Evaluate the limit of which is constant as approaches .

Step 9

Evaluate the limit of by plugging in for .

Step 10

Step 10.1

Multiply by .

Step 10.2

Add and .

Step 11

Step 11.1

Multiply by .

Step 11.2

Add and .

Step 12

The slope is and the point is .

Step 13

Step 13.1

Use the formula for the equation of a line to find .

Step 13.2

Substitute the value of into the equation.

Step 13.3

Substitute the value of into the equation.

Step 13.4

Substitute the value of into the equation.

Step 13.5

Find the value of .

Step 13.5.1

Rewrite the equation as .

Step 13.5.2

Multiply by .

Step 13.5.3

Move all terms not containing to the right side of the equation.

Step 13.5.3.1

Subtract from both sides of the equation.

Step 13.5.3.2

Subtract from .

Step 14

Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.

Step 15