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

Problem

d()/d(x)*(x−√(3,x))

Solution

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

x−√(3,x)=x−x(1/3)

2. Apply the sum rule for derivatives, which allows for the differentiation of each term independently.

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

3. Apply the power rule, which states that d(xn)/d(x)=n*x(n−1)

d(x)/d(x)=1

d(x(1/3))/d(x)=1/3*x(1/3−1)

4. Simplify the exponent in the second term.

1/3*x(−2/3)

5. Combine the results and rewrite the negative exponent as a fraction if desired.

1−1/(3*x(2/3))

Final Answer

d()/d(x)*(x−√(3,x))=1−1/(3√(3,x2))

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