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

Problem

d(√(,x))/d(x)

Solution

1. Rewrite the expression using exponent rules to simplify the base.

√(,x)=x(1/2)

2. Apply the power of a power rule to combine the exponents.

(√(,x))x=(x(1/2))x=x(x/2)

3. Use logarithmic differentiation or the identity ab=e(b*ln(a)) to rewrite the function for differentiation.

x(x/2)=eln(x(x/2))=e(x/2*ln(x))

4. Apply the chain rule, where the derivative of eu is eu⋅d(u)/d(x)

d(e(x/2*ln(x)))/d(x)=e(x/2*ln(x))⋅(d(x/2)*ln(x))/d(x)

5. Apply the product rule to differentiate the exponent x/2*ln(x)

(d(x/2)*ln(x))/d(x)=1/2*ln(x)+x/2⋅1/x

6. Simplify the result of the product rule.

1/2*ln(x)+1/2=1/2*(ln(x)+1)

7. Substitute the original expression back into the derivative formula.

d(√(,x))/d(x)=(√(,x))x⋅1/2*(ln(x)+1)

Final Answer

d(√(,x))/d(x)=((√(,x))x*(ln(x)+1))/2

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