Find the Derivative - d/dx e^(e^x)

Problem

d(e(ex))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is eu where u=ex

2. Apply the Chain Rule which states that (d(ƒ)*(g(x)))/d(x)=ƒ′*(g(x))⋅g(x)′

3. Differentiate the outer function eu with respect to u which results in eu

4. Differentiate the inner function ex with respect to x which results in ex

5. Multiply the results together and substitute u=ex back into the expression.

6. Simplify the expression using the laws of exponents if necessary, though leaving it in product form is standard.

Final Answer

d(e(ex))/d(x)=e(ex)*ex

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