Find the Derivative - d/dx ( square root of x)/(x^3+1)

Problem

d()/d(x)√(,x)/(x3+1)

Solution

1. Identify the function as a quotient of two functions, u=√(,x) and v=x3+1 which requires the quotient rule.

2. Rewrite the square root as a power to make differentiation easier: u=x(1/2)

3. Differentiate the numerator u and the denominator v individually:

d(x(1/2))/d(x)=1/2*x(−1/2)

d(x3+1)/d(x)=3*x2

4. Apply the quotient rule formula, which is (vd(u)/d(x)−ud(v)/d(x))/(v2)

((x3+1)*(1/2*x(−1/2))−(x(1/2))*(3*x2))/((x3+1)2)

5. Simplify the numerator by distributing and combining like terms:

(1/2*x(5/2)+1/2*x(−1/2)−3*x(5/2))/((x3+1)2)

(−5/2*x(5/2)+1/2*x(−1/2))/((x3+1)2)

6. Factor out common terms to clear the negative exponent and the fraction:

(1−5*x3)/(2√(,x)*(x3+1)2)

Final Answer

d()/d(x)√(,x)/(x3+1)=(1−5*x3)/(2√(,x)*(x3+1)2)

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