Find the Derivative - d/dx fourth root of x^3

Problem

d(√(4,x3))/d(x)

Solution

1. Rewrite the radical expression using a rational exponent. The fourth root of x3 is equivalent to x raised to the power of the inside exponent divided by the root index.

√(4,x3)=x(3/4)

2. Apply the power rule for differentiation, which states that d(xn)/d(x)=n*x(n−1) Here, n=3/4

d(x(3/4))/d(x)=3/4*x(3/4−1)

3. Simplify the exponent by performing the subtraction.

3/4−1=−1/4

4. Rewrite the expression with a positive exponent by moving the variable to the denominator.

3/4*x(−1/4)=3/(4*x(1/4))

5. Convert the rational exponent back into radical form.

3/(4*x(1/4))=3/(4√(4,x))

Final Answer

d(√(4,x3))/d(x)=3/(4√(4,x))

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