Math Problem Statement
differentiate (e^2x+e^x+1)/e^x
Solution
To differentiate the function , we can start by simplifying the expression.
Step 1: Simplify the Expression
We can rewrite each term in the numerator divided by :
Step 2: Differentiate the Simplified Expression
Now we need to differentiate with respect to :
- The derivative of is .
- The derivative of a constant, like 1, is 0.
- The derivative of is (using the chain rule).
Putting it all together:
Final Answer
Would you like more details on any part of this process or have any questions? Here are some related questions you might find interesting:
- How do you differentiate a product of two functions?
- What is the chain rule, and how is it applied in differentiation?
- How would you integrate the function ?
- How do exponential functions differ from polynomial functions in terms of differentiation?
- What are the applications of exponential functions in real-life problems?
Tip: Remember that simplifying an expression before differentiating often makes the differentiation process easier and less error-prone.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Exponential Functions
Chain Rule
Formulas
Differentiation rules for exponential functions
Theorems
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Suitable Grade Level
Grades 11-12