Calculus Examples

Find the Critical Points (e^x)/x
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.3
Differentiate using the Power Rule.
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Step 1.1.3.1
Differentiate using the Power Rule which states that is where .
Step 1.1.3.2
Multiply by .
Step 1.1.4
Simplify.
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Step 1.1.4.1
Reorder terms.
Step 1.1.4.2
Factor out of .
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Step 1.1.4.2.1
Factor out of .
Step 1.1.4.2.2
Factor out of .
Step 1.1.4.2.3
Factor out of .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
Set the numerator equal to zero.
Step 2.3
Solve the equation for .
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Step 2.3.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3.2
Set equal to and solve for .
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Step 2.3.2.1
Set equal to .
Step 2.3.2.2
Solve for .
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Step 2.3.2.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 2.3.2.2.2
The equation cannot be solved because is undefined.
Undefined
Step 2.3.2.2.3
There is no solution for
No solution
No solution
No solution
Step 2.3.3
Set equal to and solve for .
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Step 2.3.3.1
Set equal to .
Step 2.3.3.2
Add to both sides of the equation.
Step 2.3.4
The final solution is all the values that make true.
Step 3
Find the values where the derivative is undefined.
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Step 3.1
Set the denominator in equal to to find where the expression is undefined.
Step 3.2
Solve for .
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Step 3.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.2.2
Simplify .
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Step 3.2.2.1
Rewrite as .
Step 3.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3.2.2.3
Plus or minus is .
Step 4
Evaluate at each value where the derivative is or undefined.
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Step 4.1
Evaluate at .
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Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
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Step 4.1.2.1
Divide by .
Step 4.1.2.2
Simplify.
Step 4.2
Evaluate at .
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Step 4.2.1
Substitute for .
Step 4.2.2
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Step 4.3
List all of the points.
Step 5