Math Problem Statement

differentiate (e^2x+e^x+1)/e^x

Solution

To differentiate the function e2x+ex+1ex\frac{e^{2x} + e^x + 1}{e^x}, we can start by simplifying the expression.

Step 1: Simplify the Expression

e2x+ex+1ex\frac{e^{2x} + e^x + 1}{e^x}

We can rewrite each term in the numerator divided by exe^x:

=e2xex+exex+1ex= \frac{e^{2x}}{e^x} + \frac{e^x}{e^x} + \frac{1}{e^x}

=e2xx+exx+ex= e^{2x - x} + e^{x - x} + e^{-x}

=ex+1+ex= e^x + 1 + e^{-x}

Step 2: Differentiate the Simplified Expression

Now we need to differentiate ex+1+exe^x + 1 + e^{-x} with respect to xx:

  1. The derivative of exe^x is exe^x.
  2. The derivative of a constant, like 1, is 0.
  3. The derivative of exe^{-x} is ex-e^{-x} (using the chain rule).

Putting it all together:

ddx(ex+1+ex)=ex+0ex\frac{d}{dx}\left(e^x + 1 + e^{-x}\right) = e^x + 0 - e^{-x}

=exex= e^x - e^{-x}

Final Answer

ddx(e2x+ex+1ex)=exex\frac{d}{dx}\left(\frac{e^{2x} + e^x + 1}{e^x}\right) = e^x - e^{-x}

Would you like more details on any part of this process or have any questions? Here are some related questions you might find interesting:

  1. How do you differentiate a product of two functions?
  2. What is the chain rule, and how is it applied in differentiation?
  3. How would you integrate the function exexe^x - e^{-x}?
  4. How do exponential functions differ from polynomial functions in terms of differentiation?
  5. 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