Find the Derivative - d/dx x^(1/x)

Problem

d(x(1/x))/d(x)

Solution

1. Set the expression equal to y to prepare for logarithmic differentiation.

y=x(1/x)

2. Take the natural logarithm of both sides to move the exponent.

ln(y)=ln(x(1/x))

3. Apply the power rule for logarithms to simplify the right side.

ln(y)=1/x*ln(x)

4. Differentiate implicitly with respect to x on both sides.

d(ln(y))/d(x)=d(ln(x)/x)/d(x)

5. Apply the chain rule to the left side and the quotient rule to the right side.

1/yd(y)/d(x)=(x⋅d(ln(x))/d(x)−ln(x)⋅d(x)/d(x))/(x2)

6. Evaluate the derivatives on the right side.

1/yd(y)/d(x)=(x⋅1/x−ln(x)⋅1)/(x2)

7. Simplify the numerator.

1/yd(y)/d(x)=(1−ln(x))/(x2)

8. Solve for the derivative by multiplying both sides by y

d(y)/d(x)=y⋅(1−ln(x))/(x2)

9. Substitute the original expression for y back into the equation.

d(y)/d(x)=x(1/x)⋅(1−ln(x))/(x2)

Final Answer

d(x(1/x))/d(x)=x(1/x)*((1−ln(x))/(x2))

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