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

Problem

d()/d(x)*x(1−x)

Solution

1. Identify the function as a variable base raised to a variable power, which requires logarithmic differentiation.

2. Set the equation to y=x(1−x) and take the natural logarithm of both sides.

ln(y)=ln(x(1−x))

3. Apply the power rule for logarithms to move the exponent in front of the natural log.

ln(y)=(1−x)*ln(x)

4. Differentiate both sides with respect to x using the chain rule on the left and the product rule on the right.

1/yd(y)/d(x)=d(1−x)/d(x)*ln(x)+(1−x)d(ln(x))/d(x)

5. Compute the derivatives of the individual terms.

1/yd(y)/d(x)=−1⋅ln(x)+(1−x)⋅1/x

6. Simplify the expression on the right side.

1/yd(y)/d(x)=−ln(x)+(1−x)/x

7. Solve for d(y)/d(x) by multiplying both sides by y

d(y)/d(x)=y*((1−x)/x−ln(x))

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

d(y)/d(x)=x(1−x)*((1−x)/x−ln(x))

Final Answer

d(x(1−x))/d(x)=x(1−x)*((1−x)/x−ln(x))

Want more problems? Check here!