Find the Derivative - d/dx

Problem

d()/d(x)*(√(,x)+1/√(3,x))2

Solution

1. Rewrite the expression using rational exponents to make differentiation easier.

√(,x)=x(1/2)

1/√(3,x)=x(−1/3)

2. Expand the binomial square using the identity (a+b)2=a2+2*a*b+b2

(x(1/2)+x(−1/3))2=(x(1/2))2+2*(x(1/2))*(x(−1/3))+(x(−1/3))2

3. Simplify the expanded terms by applying exponent rules.

(x(1/2))2=x

2*(x(1/2))*(x(−1/3))=2*x(1/2−1/3)=2*x(1/6)

(x(−1/3))2=x(−2/3)

4. Apply the power rule d(xn)/d(x)=n*x(n−1) to each term of the simplified expression x+2*x(1/6)+x(−2/3)

d(x)/d(x)=1

(d(2)*x(1/6))/d(x)=2⋅1/6*x(1/6−1)=1/3*x(−5/6)

d(x(−2/3))/d(x)=−2/3*x(−2/3−1)=−2/3*x(−5/3)

5. Combine the results to find the final derivative.

d()/d(x)*(x+2*x(1/6)+x(−2/3))=1+1/3*x(−5/6)−2/3*x(−5/3)

Final Answer

d()/d(x)*(√(,x)+1/√(3,x))2=1+1/(3*x(5/6))−2/(3*x(5/3))

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