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

Evaluate .

Differentiate using the Power Rule which states that is where .

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 .

Move the negative in front of the fraction.

Evaluate .

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

Differentiate using the Power Rule which states that is where .

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 .

Move the negative in front of the fraction.

Combine and .

Combine and .

Move to the denominator using the negative exponent rule .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Rewrite the expression using the negative exponent rule .

Multiply and .

Set the derivative equal to .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

has factors of and .

Since has no factors besides and .

is a prime number

The number is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.

Multiply by .

The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.

The LCM for is the numeric part multiplied by the variable part.

Simplify each term.

Rewrite using the commutative property of multiplication.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of .

Move the leading negative in into the numerator.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Multiply .

Multiply by .

Multiply by .

Solve the equation.

Add to both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Simplify the left side of the equation by cancelling the common factors.

Cancel the common factor.

Divide by .

Divide by .

Raise each side of the equation to the power to eliminate the fractional exponent on the left side.

Raise to the power of .

Substitute the values of which cause the derivative to be into the original function.

Remove parentheses.

Set the base in greater than or equal to to find where the expression is defined.

Set the denominator in equal to to find where the expression is undefined.

Solve for .

Divide each term by and simplify.

Divide each term in by .

Simplify the left side of the equation by cancelling the common factors.

Cancel the common factor.

Divide by .

Divide by .

Raise each side of the equation to the power to eliminate the fractional exponent on the left side.

Raising to any positive power yields .

Set the denominator in equal to to find where the expression is undefined.

Solve for .

Divide each term by and simplify.

Divide each term in by .

Simplify the left side of the equation by cancelling the common factors.

Cancel the common factor.

Divide by .

Divide by .

Raise each side of the equation to the power to eliminate the fractional exponent on the left side.

Simplify .

Simplify the expression.

Rewrite as .

Apply the power rule and multiply exponents, .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Raising to any positive power yields .

The domain is all values of that make the expression defined.

Interval Notation:

Set-Builder Notation:

Interval Notation:

Set-Builder Notation:

Substitute the values of which cause the derivative to be undefined into the original equation.

Simplify each term.

Rewrite as .

Apply the power rule and multiply exponents, .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Raising to any positive power yields .

Rewrite as .

Apply the power rule and multiply exponents, .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Evaluate the exponent.

Multiply by .

Add and .

The critical points of a function are where the value of makes the derivative or undefined.

Find the Critical Points h(t)=t^(3/4)-6t^(1/4)