Find the Derivative - d/dx 3 square root of x^5

Problem

d()/d(x)*3√(,x5)

Solution

1. Rewrite the radical expression using a fractional exponent. Since √(,xn)=x(n/2) the expression becomes 3*x(5/2)

3√(,x5)=3*x(5/2)

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

d()/d(x)*3*x(5/2)=5/2⋅3*x(5/2−1)

3. Simplify the coefficient by multiplying the constant and the exponent.

5/2⋅3=15/2

4. Subtract the exponents to find the new power of x

5/2−2/2=3/2

5. Combine the results to form the final derivative expression.

15/2*x(3/2)

Final Answer

(d(3)√(,x5))/d(x)=(15√(,x3))/2

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