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

Problem

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

Solution

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

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

x(ex)=e(ex*ln(x))

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

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

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

(d(ex)*ln(x))/d(x)=d(ex)/d(x)⋅ln(x)+ex⋅d(ln(x))/d(x)

5. Compute the derivatives of the individual components.

(d(ex)*ln(x))/d(x)=ex*ln(x)+ex⋅1/x

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

d(x(ex))/d(x)=x(ex)*(ex*ln(x)+(ex)/x)

7. Factor out the common term ex

d(x(ex))/d(x)=x(ex)*ex*(ln(x)+1/x)

Final Answer

d(x(ex))/d(x)=x(ex)*ex*(ln(x)+1/x)

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