Find the Derivative - d/dx ( square root of x+x)/(x^2)

Problem

d()/d(x)(√(,x)+x)/(x2)

Solution

1. Rewrite the expression by splitting the fraction into two separate terms to simplify the differentiation process.

(√(,x)+x)/(x2)=(x(1/2))/(x2)+x/(x2)

2. Simplify the terms using the laws of exponents, specifically xa/xb=x(a−b)

x(1/2−2)+x(1−2)=x(−3/2)+x(−1)

3. Apply the power rule for derivatives, which states d(xn)/d(x)=n*x(n−1) to each term individually.

d()/d(x)*(x(−3/2)+x(−1))=−3/2*x(−5/2)−1*x(−2)

4. Rewrite the result using positive exponents and radical notation for the final form.

−3/(2*x(5/2))−1/(x2)

Final Answer

d()/d(x)(√(,x)+x)/(x2)=−3/(2*x2√(,x))−1/(x2)

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