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

Problem

d()/d(x)(√(,x)+x)/(x^2)

Solution

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

(√(,x)+x)/(x^2)=(x^(1/2))/(x^2)+x/(x^2)

2. Simplify the terms using the laws of exponents, specifically x^a/x^b=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(x^n)/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/(x^2)

Final Answer

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

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