# Calculus Examples

,

Find the first derivative.

Differentiate.

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

Differentiate using the Power Rule which states that is where .

Evaluate .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

Evaluate .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

Differentiate using the Constant Rule.

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

Add and .

The first derivative of with respect to is .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

Set-Builder Notation:

is continuous on .

is continuous

The average value of function over the interval is defined as .

Substitute the actual values into the formula for the average value of a function.

Split the single integral into multiple integrals.

Since is constant with respect to , move out of the integral.

By the Power Rule, the integral of with respect to is .

Combine and .

Since is constant with respect to , move out of the integral.

By the Power Rule, the integral of with respect to is .

Combine and .

Since is constant with respect to , move out of the integral.

Evaluate at and at .

Evaluate at and at .

Evaluate at and at .

Simplify.

Raise to the power of .

Raising to any positive power yields .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Combine and .

Multiply by .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Raise to the power of .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Raising to any positive power yields .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Multiply by .

Subtract from .

Multiply by .

Multiply by .

Add and .

Add and .

Multiply by .

Add and .

Factor out of .

Cancel the common factor.

Rewrite the expression.