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Calculus Examples
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Evaluate .
Step 1.1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.1.2.3
Combine and .
Step 1.1.2.4
Combine the numerators over the common denominator.
Step 1.1.2.5
Simplify the numerator.
Step 1.1.2.5.1
Multiply by .
Step 1.1.2.5.2
Subtract from .
Step 1.1.2.6
Move the negative in front of the fraction.
Step 1.1.3
Evaluate .
Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.3.4
Combine and .
Step 1.1.3.5
Combine the numerators over the common denominator.
Step 1.1.3.6
Simplify the numerator.
Step 1.1.3.6.1
Multiply by .
Step 1.1.3.6.2
Subtract from .
Step 1.1.3.7
Move the negative in front of the fraction.
Step 1.1.3.8
Combine and .
Step 1.1.3.9
Multiply by .
Step 1.1.3.10
Multiply by .
Step 1.1.3.11
Move to the left of .
Step 1.1.3.12
Move to the denominator using the negative exponent rule .
Step 1.1.4
Simplify.
Step 1.1.4.1
Rewrite the expression using the negative exponent rule .
Step 1.1.4.2
Multiply by .
Step 1.2
The first derivative of with respect to is .
Step 2
Step 2.1
Set the first derivative equal to .
Step 2.2
Find the LCD of the terms in the equation.
Step 2.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.2.2
Since contains 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 .
Step 2.2.3
The LCM is the smallest positive 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.
Step 2.2.4
Since has no factors besides and .
is a prime number
Step 2.2.5
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 2.2.6
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 2.2.7
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 2.2.8
The LCM for is the numeric part multiplied by the variable part.
Step 2.3
Multiply each term in by to eliminate the fractions.
Step 2.3.1
Multiply each term in by .
Step 2.3.2
Simplify the left side.
Step 2.3.2.1
Simplify each term.
Step 2.3.2.1.1
Rewrite using the commutative property of multiplication.
Step 2.3.2.1.2
Cancel the common factor of .
Step 2.3.2.1.2.1
Cancel the common factor.
Step 2.3.2.1.2.2
Rewrite the expression.
Step 2.3.2.1.3
Cancel the common factor of .
Step 2.3.2.1.3.1
Factor out of .
Step 2.3.2.1.3.2
Cancel the common factor.
Step 2.3.2.1.3.3
Rewrite the expression.
Step 2.3.2.1.4
Divide by .
Step 2.3.2.1.5
Simplify.
Step 2.3.2.1.6
Rewrite using the commutative property of multiplication.
Step 2.3.2.1.7
Cancel the common factor of .
Step 2.3.2.1.7.1
Cancel the common factor.
Step 2.3.2.1.7.2
Rewrite the expression.
Step 2.3.2.1.8
Cancel the common factor of .
Step 2.3.2.1.8.1
Cancel the common factor.
Step 2.3.2.1.8.2
Rewrite the expression.
Step 2.3.3
Simplify the right side.
Step 2.3.3.1
Multiply .
Step 2.3.3.1.1
Multiply by .
Step 2.3.3.1.2
Multiply by .
Step 2.4
Subtract from both sides of the equation.
Step 3
Step 3.1
Convert expressions with fractional exponents to radicals.
Step 3.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 3.1.2
Apply the rule to rewrite the exponentiation as a radical.
Step 3.2
Set the denominator in equal to to find where the expression is undefined.
Step 3.3
Solve for .
Step 3.3.1
To remove the radical on the left side of the equation, raise both sides of the equation to the power of .
Step 3.3.2
Simplify each side of the equation.
Step 3.3.2.1
Use to rewrite as .
Step 3.3.2.2
Simplify the left side.
Step 3.3.2.2.1
Simplify .
Step 3.3.2.2.1.1
Apply the product rule to .
Step 3.3.2.2.1.2
Raise to the power of .
Step 3.3.2.2.1.3
Multiply the exponents in .
Step 3.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 3.3.2.2.1.3.2
Cancel the common factor of .
Step 3.3.2.2.1.3.2.1
Cancel the common factor.
Step 3.3.2.2.1.3.2.2
Rewrite the expression.
Step 3.3.2.3
Simplify the right side.
Step 3.3.2.3.1
Raising to any positive power yields .
Step 3.3.3
Solve for .
Step 3.3.3.1
Divide each term in by and simplify.
Step 3.3.3.1.1
Divide each term in by .
Step 3.3.3.1.2
Simplify the left side.
Step 3.3.3.1.2.1
Cancel the common factor of .
Step 3.3.3.1.2.1.1
Cancel the common factor.
Step 3.3.3.1.2.1.2
Divide by .
Step 3.3.3.1.3
Simplify the right side.
Step 3.3.3.1.3.1
Divide by .
Step 3.3.3.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.3.3.3
Simplify .
Step 3.3.3.3.1
Rewrite as .
Step 3.3.3.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3.3.3.3.3
Plus or minus is .
Step 3.4
Set the denominator in equal to to find where the expression is undefined.
Step 3.5
Solve for .
Step 3.5.1
To remove the radical on the left side of the equation, raise both sides of the equation to the power of .
Step 3.5.2
Simplify each side of the equation.
Step 3.5.2.1
Use to rewrite as .
Step 3.5.2.2
Simplify the left side.
Step 3.5.2.2.1
Simplify .
Step 3.5.2.2.1.1
Apply the product rule to .
Step 3.5.2.2.1.2
Raise to the power of .
Step 3.5.2.2.1.3
Multiply the exponents in .
Step 3.5.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 3.5.2.2.1.3.2
Cancel the common factor of .
Step 3.5.2.2.1.3.2.1
Cancel the common factor.
Step 3.5.2.2.1.3.2.2
Rewrite the expression.
Step 3.5.2.3
Simplify the right side.
Step 3.5.2.3.1
Raising to any positive power yields .
Step 3.5.3
Solve for .
Step 3.5.3.1
Divide each term in by and simplify.
Step 3.5.3.1.1
Divide each term in by .
Step 3.5.3.1.2
Simplify the left side.
Step 3.5.3.1.2.1
Cancel the common factor of .
Step 3.5.3.1.2.1.1
Cancel the common factor.
Step 3.5.3.1.2.1.2
Divide by .
Step 3.5.3.1.3
Simplify the right side.
Step 3.5.3.1.3.1
Divide by .
Step 3.5.3.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.5.3.3
Simplify .
Step 3.5.3.3.1
Rewrite as .
Step 3.5.3.3.2
Pull terms out from under the radical, assuming real numbers.
Step 4
Step 4.1
Evaluate at .
Step 4.1.1
Substitute for .
Step 4.1.2
Rewrite the expression using the negative exponent rule .
Step 4.2
Evaluate at .
Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
Step 4.2.2.1
Simplify each term.
Step 4.2.2.1.1
Rewrite as .
Step 4.2.2.1.2
Apply the power rule and multiply exponents, .
Step 4.2.2.1.3
Cancel the common factor of .
Step 4.2.2.1.3.1
Cancel the common factor.
Step 4.2.2.1.3.2
Rewrite the expression.
Step 4.2.2.1.4
Evaluate the exponent.
Step 4.2.2.1.5
Rewrite the expression using the negative exponent rule .
Step 4.2.2.1.6
Simplify the denominator.
Step 4.2.2.1.6.1
Rewrite as .
Step 4.2.2.1.6.2
Apply the power rule and multiply exponents, .
Step 4.2.2.1.6.3
Cancel the common factor of .
Step 4.2.2.1.6.3.1
Cancel the common factor.
Step 4.2.2.1.6.3.2
Rewrite the expression.
Step 4.2.2.1.6.4
Raising to any positive power yields .
Step 4.2.2.1.6.5
The expression contains a division by . The expression is undefined.
Undefined
Step 4.2.2.1.7
The expression contains a division by . The expression is undefined.
Undefined
Step 4.2.2.2
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Undefined
Step 4.3
List all of the points.
Step 5