# Calculus Examples

Step 1

Step 1.1

By the Sum Rule, the derivative of with respect to is .

Step 1.2

Evaluate .

Step 1.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.2.2

Differentiate using the Power Rule which states that is where .

Step 1.2.3

Multiply by .

Step 1.3

Evaluate .

Step 1.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.3.2

Differentiate using the Power Rule which states that is where .

Step 1.3.3

Multiply by .

Step 1.4

Differentiate.

Step 1.4.1

Differentiate using the Power Rule which states that is where .

Step 1.4.2

Since is constant with respect to , the derivative of with respect to is .

Step 1.5

Simplify.

Step 1.5.1

Add and .

Step 1.5.2

Reorder terms.

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

Step 3

Split into separate intervals around the values that make the first derivative or undefined.

Step 4

Step 4.1

Replace the variable with in the expression.

Step 4.2

Simplify the result.

Step 4.2.1

Simplify each term.

Step 4.2.1.1

Raising to any positive power yields .

Step 4.2.1.2

Multiply by .

Step 4.2.1.3

Raising to any positive power yields .

Step 4.2.1.4

Multiply by .

Step 4.2.2

Simplify by adding numbers.

Step 4.2.2.1

Add and .

Step 4.2.2.2

Add and .

Step 4.2.3

The final answer is .

Step 5

Step 5.1

Replace the variable with in the expression.

Step 5.2

Simplify the result.

Step 5.2.1

Simplify each term.

Step 5.2.1.1

Raise to the power of .

Step 5.2.1.2

Multiply by .

Step 5.2.1.3

Raise to the power of .

Step 5.2.1.4

Multiply by .

Step 5.2.2

Simplify by adding numbers.

Step 5.2.2.1

Add and .

Step 5.2.2.2

Add and .

Step 5.2.3

The final answer is .

Step 6

Since the first derivative changed signs from positive to negative around , then there is a turning point at .

Step 7

Step 7.1

Find to find the y-coordinate of .

Step 7.1.1

Replace the variable with in the expression.

Step 7.1.2

Simplify .

Step 7.1.2.1

Remove parentheses.

Step 7.1.2.2

Simplify each term.

Step 7.1.2.2.1

Raise to the power of .

Step 7.1.2.2.2

Multiply by .

Step 7.1.2.2.3

Raise to the power of .

Step 7.1.2.2.4

Multiply by .

Step 7.1.2.3

Simplify by adding and subtracting.

Step 7.1.2.3.1

Subtract from .

Step 7.1.2.3.2

Add and .

Step 7.1.2.3.3

Add and .

Step 7.2

Write the and coordinates in point form.

Step 8