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

Problem

d(√(5,x))/d(x)

Solution

1. Rewrite the radical expression using a rational exponent to make it easier to apply the power rule.

√(5,x)=x(1/5)

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

d(x(1/5))/d(x)=1/5*x(1/5−1)

3. Subtract the exponents to simplify the expression.

1/5−1=−4/5

4. Simplify the result by moving the negative exponent to the denominator and converting back to radical form if desired.

1/5*x(−4/5)=1/(5*x(4/5))

Final Answer

d(√(5,x))/d(x)=1/(5√(5,x4))

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