Find the Derivative - d/dx 1/( cube root of x^2)

Problem

d()/d(x)1/√(3,x2)

Solution

1. Rewrite the expression using rational exponents to make it easier to differentiate.

1/√(3,x2)=1/(x(2/3))

2. Apply the rule for negative exponents to move the term to the numerator.

1/(x(2/3))=x(−2/3)

3. Use the power rule for differentiation, which states that d(xn)/d(x)=n*x(n−1)

d(x(−2/3))/d(x)=−2/3*x(−2/3−1)

4. Simplify the exponent by performing the subtraction.

−2/3*x(−5/3)

5. Convert the expression back into radical form with a positive exponent.

−2/(3*x(5/3))=−2/(3√(3,x5))

Final Answer

d()/d(x)1/√(3,x2)=−2/(3√(3,x5))

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