Find the Derivative - d/dx y=( square root of x)^x

Problem

d()/d(x)*(√(,x))x

Solution

1. Rewrite the function using the property of exponents √(,x)=x(1/2) to simplify the base.

y=(x(1/2))x

2. Simplify the expression by multiplying the exponents using the power of a power rule (am)n=a(m*n)

y=x(x/2)

3. Apply logarithmic differentiation by taking the natural logarithm of both sides to handle the variable in the exponent.

ln(y)=ln(x(x/2))

4. Use the logarithm power rule ln(ab)=b*ln(a) to move the exponent in front of the logarithm.

ln(y)=x/2*ln(x)

5. 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()/d(x)*(x/2)⋅ln(x)+x/2⋅d(ln(x))/d(x)

6. Compute the derivatives of the individual terms on the right side.

1/yd(y)/d(x)=1/2*ln(x)+x/2⋅1/x

7. Simplify the resulting expression on the right side.

1/yd(y)/d(x)=ln(x)/2+1/2

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

d(y)/d(x)=y((ln(x)+1)/2)

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

d(y)/d(x)=(√(,x))x*((ln(x)+1)/2)

Final Answer

d(√(,x))/d(x)=(√(,x))x*((ln(x)+1)/2)

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