# Calculus Examples

Find the Average Value of the Derivative
,
Finding the derivative of .
Find the first derivative.
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 .
Since is constant with respect to , the derivative of with respect to is .
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.
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Raise to the power of .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Combine and .
Multiply by .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Multiply by .