Find dy/dx y=x^( square root of x)

Problem

d()/d(x)*x√(,x)

Solution

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

2. Take the natural log of both sides to simplify the exponent using the property ln(ab)=b*ln(a)

ln(y)=ln(x√(,x))

ln(y)=√(,x)*ln(x)

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

d(ln(y))/d(x)=(d(√(,x))*ln(x))/d(x)

4. Apply the chain rule to the left side and the product rule to the right side.

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

5. Calculate the derivatives of the individual terms.

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

6. Simplify the expression on the right side.

1/yd(y)/d(x)=ln(x)/(2√(,x))+1/√(,x)

1/yd(y)/d(x)=(ln(x)+2)/(2√(,x))

7. Solve for dy/dx by multiplying both sides by y

d(y)/d(x)=y(ln(x)+2)/(2√(,x))

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

d(y)/d(x)=x√(,x)(ln(x)+2)/(2√(,x))

Final Answer

d(x√(,x))/d(x)=x√(,x)(ln(x)+2)/(2√(,x))

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