Enter a problem...
Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
Find the first derivative.
Step 2.1.1
Differentiate using the Product Rule which states that is where and .
Step 2.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.1.3
Differentiate.
Step 2.1.3.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.3
Add and .
Step 2.1.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.5
Differentiate using the Power Rule which states that is where .
Step 2.1.3.6
Simplify the expression.
Step 2.1.3.6.1
Multiply by .
Step 2.1.3.6.2
Move to the left of .
Step 2.1.4
Simplify.
Step 2.1.4.1
Apply the distributive property.
Step 2.1.4.2
Combine terms.
Step 2.1.4.2.1
Subtract from .
Step 2.1.4.2.2
Add and .
Step 2.1.4.3
Reorder the factors of .
Step 2.1.4.4
Reorder factors in .
Step 2.2
The first derivative of with respect to is .
Step 3
Step 3.1
Set the first derivative equal to .
Step 3.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.3
Set equal to .
Step 3.4
Set equal to and solve for .
Step 3.4.1
Set equal to .
Step 3.4.2
Solve for .
Step 3.4.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 3.4.2.2
The equation cannot be solved because is undefined.
Undefined
Step 3.4.2.3
There is no solution for
No solution
No solution
No solution
Step 3.5
The final solution is all the values that make true.
Step 4
The values which make the derivative equal to are .
Step 5
After finding the point that makes the derivative equal to or undefined, the interval to check where is increasing and where it is decreasing is .
Step 6
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
Multiply by .
Step 6.2.2
Rewrite the expression using the negative exponent rule .
Step 6.2.3
Combine and .
Step 6.2.4
The final answer is .
Step 6.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 7
Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
Step 7.2.1
Multiply by .
Step 7.2.2
The final answer is .
Step 7.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 8
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 9