Find the Derivative - d/dx fifth root of x^2

Problem

d()/d(x)√(5,x^2)

Solution

1. Rewrite the radical expression using a rational exponent by applying the rule √(n,x^m)=x^(m/n)

√(5,x^2)=x^(2/5)

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

d(x^(2/5))/d(x)=2/5*x^(2/5−1)

3. Subtract the exponents to simplify the expression.

2/5*x^(−3/5)

4. Rewrite the expression with a positive exponent using the rule x^(−n)=1/(x^n)

2/(5*x^(3/5))

5. Convert the rational exponent back into radical form using the rule x^(m/n)=√(n,x^m)

2/(5√(5,x^3))

Final Answer

d(√(5,x^2))/d(x)=2/(5√(5,x^3))

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