Calculus Examples

Find the first derivative of the function.
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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 .
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 .
Multiply by .
Find the second derivative of the function.
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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 .
Multiply by .
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 .
Multiply by .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
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 .
Set equal to and solve for .
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Set the factor equal to .
Add to both sides of the equation.
Take the root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
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First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
The solution is the result of and .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Raising to any positive power yields .
Multiply by .
Subtract from .
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Find the y-value when .
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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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Raising to any positive power yields .
Raising to any positive power yields .
Multiply by .
Add and .
The final answer is .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Find the y-value when .
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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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Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
The final answer is .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Find the y-value when .
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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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Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
The final answer is .
These are the local extrema for .
is a local maxima
is a local minima
is a local minima
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