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

Problem

d(x^(e^x))/d(x)

Solution

1. Identify the function as a variable base raised to a variable power, which requires logarithmic differentiation or the identity x^(e^x)=e^ln(x^(e^x))

2. Rewrite the expression using the exponential identity to prepare for the chain rule.

x^(e^x)=e^(e^x*ln(x))

3. Apply the chain rule for the derivative of e^u where u=e^x*ln(x)

d(e^(e^x*ln(x)))/d(x)=e^(e^x*ln(x))⋅(d(e^x)*ln(x))/d(x)

4. Apply the product rule to differentiate e^x*ln(x)

(d(e^x)*ln(x))/d(x)=d(e^x)/d(x)⋅ln(x)+e^x⋅d(ln(x))/d(x)

5. Compute the derivatives of the individual components.

(d(e^x)*ln(x))/d(x)=e^x*ln(x)+e^x⋅1/x

6. Substitute the result back into the chain rule expression and simplify by replacing e^(e^x*ln(x)) with the original x^(e^x)

d(x^(e^x))/d(x)=x^(e^x)*(e^x*ln(x)+(e^x)/x)

7. Factor out the common term e^x

d(x^(e^x))/d(x)=x^(e^x)*e^x*(ln(x)+1/x)

Final Answer

d(x^(e^x))/d(x)=x^(e^x)*e^x*(ln(x)+1/x)

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