Calculus Examples

Find the Inflection Points x^(1/5)(x+6)
Write as a function.
Find the second derivative.
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Find the first derivative.
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Differentiate using the Product Rule which states that is where and .
Differentiate.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Simplify the expression.
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Add and .
Multiply by .
Differentiate using the Power Rule which states that is where .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Move to the denominator using the negative exponent rule .
Simplify.
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Apply the distributive property.
Combine terms.
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Combine and .
Move to the numerator using the negative exponent rule .
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Subtract from .
Combine and .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Move to the left of .
Add and .
Find the second derivative.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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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 .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Multiply and .
Multiply by .
Move to the denominator using the negative exponent rule .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Rewrite as .
Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate using the Power Rule which states that is where .
Multiply the exponents in .
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Apply the power rule and multiply exponents, .
Multiply .
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Combine and .
Multiply by .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Combine and .
Multiply by by adding the exponents.
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Move .
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Subtract from .
Move the negative in front of the fraction.
Move to the denominator using the negative exponent rule .
Multiply and .
Multiply by .
Multiply by .
The second derivative of with respect to is .
Set the second derivative equal to then solve the equation .
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Set the second derivative equal to .
Find the LCD of the terms in the equation.
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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 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.
has factors of and .
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.
Multiply each term by and simplify.
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Multiply each term in by in order to remove all the denominators from the equation.
Simplify each term.
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Rewrite using the commutative property of multiplication.
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Simplify.
Cancel the common factor of .
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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Multiply .
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Multiply by .
Multiply by .
Solve the equation.
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Add to both sides of the equation.
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Divide by .
Find the points where the second derivative is .
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Substitute in to find the value of .
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Replace the variable with in the expression.
Simplify the result.
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Add and .
Move to the left of .
The final answer is .
The point found by substituting in is . This point can be an inflection point.
Split into intervals around the points that could potentially be inflection points.
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of .
Multiply by .
Divide by .
Raise to the power of .
Multiply by .
Divide by .
Multiply by .
Subtract from .
The final answer is .
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of .
Multiply by .
Divide by .
Raise to the power of .
Multiply by .
Divide by .
Multiply by .
Subtract from .
The final answer is .
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is .
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