Find the Derivative - d/dz cube root of z

Problem

d(√(3,z))/d(z)

Solution

1. Rewrite the radical expression as a power with a fractional exponent.

√(3,z)=z(1/3)

2. Apply the power rule for differentiation, which states that d(zn)/d(z)=n*z(n−1)

d(z(1/3))/d(z)=1/3*z(1/3−1)

3. Subtract the exponents to simplify the expression.

1/3*z(1/3−3/3)=1/3*z(−2/3)

4. Rewrite the expression using positive exponents and radical notation if desired.

1/3*z(−2/3)=1/(3*z(2/3))=1/(3√(3,z2))

Final Answer

d(√(3,z))/d(z)=1/(3√(3,z2))

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