# Find Where Increasing/Decreasing e^(5x)+e^(-x) Find the derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
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 .
Move to the left of .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
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 .
Move to the left of .
Rewrite as .
Set the derivative equal to .
Graph each side of the equation. The solution is the x-value of the point of intersection.
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 .
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Multiply by .
Rewrite the expression using the negative exponent rule .
Combine and .
Multiply by .
The final answer is .
Simplify.
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Multiply by .
Multiply by .
Rewrite the expression using the negative exponent rule .
The final answer is .
Simplify.
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
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