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

Problem

d()/d(x)1/√(3,x)

Solution

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

1/√(3,x)=1/(x(1/3))

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

1/(x(1/3))=x(−1/3)

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

d(x(−1/3))/d(x)=−1/3*x(−1/3−1)

4. Simplify the exponent by performing the subtraction.

−1/3*x(−4/3)

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

−1/(3*x(4/3))=−1/(3√(3,x4))

Final Answer

d()/d(x)1/√(3,x)=−1/(3√(3,x4))

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