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

Problem

d(√(3,x))/d(x)

Solution

1. Rewrite the cube root as a power using the rule √(n,x)=x1/n

√(3,x)=x1/3

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

d(x1/3)/d(x)=1/3*x(1/3−1)

3. Subtract the exponents to simplify the expression.

1/3−1=−2/3

4. Simplify the result by moving the negative exponent to the denominator.

1/3*x(−2/3)=1/(3*x2/3)

5. Convert the rational exponent back into radical form if desired.

1/(3*x2/3)=1/(3√(3,x2))

Final Answer

d(√(3,x))/d(x)=1/(3√(3,x2))

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