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

Problem

d(√(,x3))/d(x)

Solution

1. Rewrite the radical expression using a fractional exponent. Since √(,a)=a(1/2) the expression becomes (x3)(1/2)

2. Simplify the exponent using the power of a power rule, which states (xa)b=x(a*b)

x(3⋅1/2)=x(3/2)

3. Apply the power rule for derivatives, d(xn)/d(x)=n*x(n−1) where n=3/2

d(x(3/2))/d(x)=3/2*x(3/2−1)

4. Subtract the exponents to find the final power.

3/2−1=1/2

5. Convert the result back into radical form if desired, noting that x(1/2)=√(,x)

3/2*x(1/2)=(3√(,x))/2

Final Answer

d(√(,x3))/d(x)=(3√(,x))/2

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