Find the Derivative - d/dx cube root of 8/x

Problem

d()/d(x)√(3,8/x)

Solution

1. Rewrite the expression using exponent notation to make it easier to differentiate.

√(3,8/x)=(8/x)(1/3)

2. Simplify the expression by applying the exponent to both the numerator and the denominator.

(8(1/3))/(x(1/3))=2/(x(1/3))

3. Express the term with a negative exponent to prepare for the power rule.

2*x(−1/3)

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

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

5. Simplify the coefficient and the exponent.

−2/3*x(−4/3)

6. Rewrite the result using positive exponents and radical notation if desired.

−2/(3*x(4/3))

Final Answer

d()/d(x)√(3,8/x)=−2/(3*x√(3,x))

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