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

Problem

d()/d(x)1/√(5,x)

Solution

1. Rewrite the radical expression using a rational exponent.

√(5,x)=x(1/5)

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

1/(x(1/5))=x(−1/5)

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

d(x(−1/5))/d(x)=−1/5*x(−1/5−1)

4. Simplify the exponent by performing the subtraction.

−1/5−5/5=−6/5

5. Rewrite the expression using positive exponents and radical notation.

−1/5*x(−6/5)=−1/(5*x(6/5))

Final Answer

d()/d(x)1/√(5,x)=−1/(5*x(6/5))

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