# 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

Remove parentheses.

Step 2.1.2.2

Simplify each term.

Step 2.1.2.2.1

One to any power is one.

Step 2.1.2.2.2

Multiply by .

Step 2.1.2.3

Simplify by adding numbers.

Step 2.1.2.3.1

Add and .

Step 2.1.2.3.2

Add and .

Step 2.1.2.4

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

Remove parentheses.

Step 5.1.2.2

Simplify each term.

Step 5.1.2.2.1

Use the Binomial Theorem.

Step 5.1.2.2.2

Apply the distributive property.

Step 5.1.2.2.3

Simplify.

Step 5.1.2.2.3.1

Multiply by .

Step 5.1.2.2.3.2

Multiply by .

Step 5.1.2.2.4

Remove parentheses.

Step 5.1.2.3

The final answer is .

Step 5.2

Reorder.

Step 5.2.1

Move .

Step 5.2.2

Move .

Step 5.2.3

Move .

Step 5.2.4

Move .

Step 5.2.5

Move .

Step 5.2.6

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

Simplify.

Step 7.1.2.1

Multiply by .

Step 7.1.2.2

Multiply by .

Step 7.1.3

Subtract from .

Step 7.1.4

Add and .

Step 7.1.5

Subtract from .

Step 7.1.6

Add and .

Step 7.1.7

Subtract from .

Step 7.1.8

Add and .

Step 7.1.9

Factor out of .

Step 7.1.9.1

Factor out of .

Step 7.1.9.2

Factor out of .

Step 7.1.9.3

Factor out of .

Step 7.1.9.4

Raise to the power of .

Step 7.1.9.5

Factor out of .

Step 7.1.9.6

Factor out of .

Step 7.1.9.7

Factor out of .

Step 7.1.9.8

Factor out of .

Step 7.2

Reduce the expression by cancelling the common factors.

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

Simplify the expression.

Step 7.2.2.1

Move .

Step 7.2.2.2

Move .

Step 7.2.2.3

Reorder and .

Step 8

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

Step 9

Evaluate the limit of which is constant as approaches .

Step 10

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

Step 11

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

Step 12

Move the exponent from outside the limit using the Limits Power Rule.

Step 13

Evaluate the limit of which is constant as approaches .

Step 14

Step 14.1

Evaluate the limit of by plugging in for .

Step 14.2

Evaluate the limit of by plugging in for .

Step 15

Step 15.1

Simplify each term.

Step 15.1.1

Multiply .

Step 15.1.1.1

Multiply by .

Step 15.1.1.2

Multiply by .

Step 15.1.2

Raising to any positive power yields .

Step 15.1.3

Multiply by .

Step 15.2

Combine the opposite terms in .

Step 15.2.1

Add and .

Step 15.2.2

Add and .

Step 16

Step 16.1

Remove parentheses.

Step 16.2

Simplify .

Step 16.2.1

Simplify each term.

Step 16.2.1.1

One to any power is one.

Step 16.2.1.2

Multiply by .

Step 16.2.2

Add and .

Step 17

The slope is and the point is .

Step 18

Step 18.1

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

Step 18.2

Substitute the value of into the equation.

Step 18.3

Substitute the value of into the equation.

Step 18.4

Substitute the value of into the equation.

Step 18.5

Find the value of .

Step 18.5.1

Rewrite the equation as .

Step 18.5.2

Multiply by .

Step 18.5.3

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

Step 18.5.3.1

Subtract from both sides of the equation.

Step 18.5.3.2

Subtract from .

Step 19

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

Step 20