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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
Find the first derivative.
Step 2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2
Evaluate .
Step 2.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2.3
Multiply by .
Step 2.1.3
Evaluate .
Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3.3
Multiply by .
Step 2.1.4
Differentiate using the Constant Rule.
Step 2.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.4.2
Add and .
Step 2.2
Find the second derivative.
Step 2.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.2
Evaluate .
Step 2.2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.2.3
Multiply by .
Step 2.2.3
Evaluate .
Step 2.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3.3
Multiply by .
Step 2.3
The second derivative of with respect to is .
Step 3
Step 3.1
Set the second derivative equal to .
Step 3.2
Factor out of .
Step 3.2.1
Factor out of .
Step 3.2.2
Factor out of .
Step 3.2.3
Factor out of .
Step 3.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.4
Set equal to .
Step 3.5
Set equal to and solve for .
Step 3.5.1
Set equal to .
Step 3.5.2
Solve for .
Step 3.5.2.1
Add to both sides of the equation.
Step 3.5.2.2
Divide each term in by and simplify.
Step 3.5.2.2.1
Divide each term in by .
Step 3.5.2.2.2
Simplify the left side.
Step 3.5.2.2.2.1
Cancel the common factor of .
Step 3.5.2.2.2.1.1
Cancel the common factor.
Step 3.5.2.2.2.1.2
Divide by .
Step 3.5.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.5.2.4
Simplify .
Step 3.5.2.4.1
Rewrite as .
Step 3.5.2.4.2
Any root of is .
Step 3.5.2.4.3
Multiply by .
Step 3.5.2.4.4
Combine and simplify the denominator.
Step 3.5.2.4.4.1
Multiply by .
Step 3.5.2.4.4.2
Raise to the power of .
Step 3.5.2.4.4.3
Raise to the power of .
Step 3.5.2.4.4.4
Use the power rule to combine exponents.
Step 3.5.2.4.4.5
Add and .
Step 3.5.2.4.4.6
Rewrite as .
Step 3.5.2.4.4.6.1
Use to rewrite as .
Step 3.5.2.4.4.6.2
Apply the power rule and multiply exponents, .
Step 3.5.2.4.4.6.3
Combine and .
Step 3.5.2.4.4.6.4
Cancel the common factor of .
Step 3.5.2.4.4.6.4.1
Cancel the common factor.
Step 3.5.2.4.4.6.4.2
Rewrite the expression.
Step 3.5.2.4.4.6.5
Evaluate the exponent.
Step 3.5.2.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 3.5.2.5.1
First, use the positive value of the to find the first solution.
Step 3.5.2.5.2
Next, use the negative value of the to find the second solution.
Step 3.5.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 3.6
The final solution is all the values that make true.
Step 4
Step 4.1
Substitute in to find the value of .
Step 4.1.1
Replace the variable with in the expression.
Step 4.1.2
Simplify the result.
Step 4.1.2.1
Simplify each term.
Step 4.1.2.1.1
Raising to any positive power yields .
Step 4.1.2.1.2
Multiply by .
Step 4.1.2.1.3
Raising to any positive power yields .
Step 4.1.2.1.4
Multiply by .
Step 4.1.2.2
Simplify by adding numbers.
Step 4.1.2.2.1
Add and .
Step 4.1.2.2.2
Add and .
Step 4.1.2.3
The final answer is .
Step 4.2
The point found by substituting in is . This point can be an inflection point.
Step 4.3
Substitute in to find the value of .
Step 4.3.1
Replace the variable with in the expression.
Step 4.3.2
Simplify the result.
Step 4.3.2.1
Simplify each term.
Step 4.3.2.1.1
Apply the product rule to .
Step 4.3.2.1.2
Simplify the numerator.
Step 4.3.2.1.2.1
Rewrite as .
Step 4.3.2.1.2.2
Raise to the power of .
Step 4.3.2.1.2.3
Rewrite as .
Step 4.3.2.1.2.3.1
Factor out of .
Step 4.3.2.1.2.3.2
Rewrite as .
Step 4.3.2.1.2.4
Pull terms out from under the radical.
Step 4.3.2.1.3
Raise to the power of .
Step 4.3.2.1.4
Cancel the common factor of and .
Step 4.3.2.1.4.1
Factor out of .
Step 4.3.2.1.4.2
Cancel the common factors.
Step 4.3.2.1.4.2.1
Factor out of .
Step 4.3.2.1.4.2.2
Cancel the common factor.
Step 4.3.2.1.4.2.3
Rewrite the expression.
Step 4.3.2.1.5
Combine and .
Step 4.3.2.1.6
Apply the product rule to .
Step 4.3.2.1.7
Simplify the numerator.
Step 4.3.2.1.7.1
Rewrite as .
Step 4.3.2.1.7.2
Raise to the power of .
Step 4.3.2.1.7.3
Rewrite as .
Step 4.3.2.1.7.3.1
Factor out of .
Step 4.3.2.1.7.3.2
Rewrite as .
Step 4.3.2.1.7.4
Pull terms out from under the radical.
Step 4.3.2.1.8
Raise to the power of .
Step 4.3.2.1.9
Cancel the common factor of and .
Step 4.3.2.1.9.1
Factor out of .
Step 4.3.2.1.9.2
Cancel the common factors.
Step 4.3.2.1.9.2.1
Factor out of .
Step 4.3.2.1.9.2.2
Cancel the common factor.
Step 4.3.2.1.9.2.3
Rewrite the expression.
Step 4.3.2.1.10
Combine and .
Step 4.3.2.1.11
Move the negative in front of the fraction.
Step 4.3.2.2
To write as a fraction with a common denominator, multiply by .
Step 4.3.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 4.3.2.3.1
Multiply by .
Step 4.3.2.3.2
Multiply by .
Step 4.3.2.4
Combine the numerators over the common denominator.
Step 4.3.2.5
Simplify each term.
Step 4.3.2.5.1
Simplify the numerator.
Step 4.3.2.5.1.1
Multiply by .
Step 4.3.2.5.1.2
Subtract from .
Step 4.3.2.5.2
Move the negative in front of the fraction.
Step 4.3.2.6
The final answer is .
Step 4.4
The point found by substituting in is . This point can be an inflection point.
Step 4.5
Substitute in to find the value of .
Step 4.5.1
Replace the variable with in the expression.
Step 4.5.2
Simplify the result.
Step 4.5.2.1
Simplify each term.
Step 4.5.2.1.1
Use the power rule to distribute the exponent.
Step 4.5.2.1.1.1
Apply the product rule to .
Step 4.5.2.1.1.2
Apply the product rule to .
Step 4.5.2.1.2
Raise to the power of .
Step 4.5.2.1.3
Simplify the numerator.
Step 4.5.2.1.3.1
Rewrite as .
Step 4.5.2.1.3.2
Raise to the power of .
Step 4.5.2.1.3.3
Rewrite as .
Step 4.5.2.1.3.3.1
Factor out of .
Step 4.5.2.1.3.3.2
Rewrite as .
Step 4.5.2.1.3.4
Pull terms out from under the radical.
Step 4.5.2.1.4
Raise to the power of .
Step 4.5.2.1.5
Cancel the common factor of and .
Step 4.5.2.1.5.1
Factor out of .
Step 4.5.2.1.5.2
Cancel the common factors.
Step 4.5.2.1.5.2.1
Factor out of .
Step 4.5.2.1.5.2.2
Cancel the common factor.
Step 4.5.2.1.5.2.3
Rewrite the expression.
Step 4.5.2.1.6
Multiply .
Step 4.5.2.1.6.1
Multiply by .
Step 4.5.2.1.6.2
Combine and .
Step 4.5.2.1.7
Move the negative in front of the fraction.
Step 4.5.2.1.8
Use the power rule to distribute the exponent.
Step 4.5.2.1.8.1
Apply the product rule to .
Step 4.5.2.1.8.2
Apply the product rule to .
Step 4.5.2.1.9
Raise to the power of .
Step 4.5.2.1.10
Simplify the numerator.
Step 4.5.2.1.10.1
Rewrite as .
Step 4.5.2.1.10.2
Raise to the power of .
Step 4.5.2.1.10.3
Rewrite as .
Step 4.5.2.1.10.3.1
Factor out of .
Step 4.5.2.1.10.3.2
Rewrite as .
Step 4.5.2.1.10.4
Pull terms out from under the radical.
Step 4.5.2.1.11
Raise to the power of .
Step 4.5.2.1.12
Cancel the common factor of and .
Step 4.5.2.1.12.1
Factor out of .
Step 4.5.2.1.12.2
Cancel the common factors.
Step 4.5.2.1.12.2.1
Factor out of .
Step 4.5.2.1.12.2.2
Cancel the common factor.
Step 4.5.2.1.12.2.3
Rewrite the expression.
Step 4.5.2.1.13
Multiply .
Step 4.5.2.1.13.1
Multiply by .
Step 4.5.2.1.13.2
Combine and .
Step 4.5.2.2
To write as a fraction with a common denominator, multiply by .
Step 4.5.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 4.5.2.3.1
Multiply by .
Step 4.5.2.3.2
Multiply by .
Step 4.5.2.4
Combine the numerators over the common denominator.
Step 4.5.2.5
Simplify the numerator.
Step 4.5.2.5.1
Multiply by .
Step 4.5.2.5.2
Add and .
Step 4.5.2.6
The final answer is .
Step 4.6
The point found by substituting in is . This point can be an inflection point.
Step 4.7
Determine the points that could be inflection points.
Step 5
Split into intervals around the points that could potentially be inflection points.
Step 6
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
Simplify each term.
Step 6.2.1.1
Raise to the power of .
Step 6.2.1.2
Multiply by .
Step 6.2.1.3
Multiply by .
Step 6.2.2
Add and .
Step 6.2.3
The final answer is .
Step 6.3
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Step 7
Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
Step 7.2.1
Simplify each term.
Step 7.2.1.1
Raise to the power of .
Step 7.2.1.2
Multiply by .
Step 7.2.1.3
Multiply by .
Step 7.2.2
Add and .
Step 7.2.3
The final answer is .
Step 7.3
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
Step 8
Step 8.1
Replace the variable with in the expression.
Step 8.2
Simplify the result.
Step 8.2.1
Simplify each term.
Step 8.2.1.1
Raise to the power of .
Step 8.2.1.2
Multiply by .
Step 8.2.1.3
Multiply by .
Step 8.2.2
Subtract from .
Step 8.2.3
The final answer is .
Step 8.3
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Step 9
Step 9.1
Replace the variable with in the expression.
Step 9.2
Simplify the result.
Step 9.2.1
Simplify each term.
Step 9.2.1.1
Raise to the power of .
Step 9.2.1.2
Multiply by .
Step 9.2.1.3
Multiply by .
Step 9.2.2
Subtract from .
Step 9.2.3
The final answer is .
Step 9.3
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
Step 10
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 points in this case are .
Step 11