Find the Derivative Using Quotient Rule - d/dx x/(x^2+1)

Problem

d()/d(x)x/(x2+1)

Solution

1. Identify the numerator and denominator for the quotient rule. Let u=x and v=x2+1

2. Differentiate both parts with respect to x We find d(u)/d(x)=1 and d(v)/d(x)=2*x

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

4. Substitute the expressions into the formula:

((x2+1)*(1)−(x)*(2*x))/((x2+1)2)

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

(x2+1−2*x2)/((x2+1)2)

6. Combine the x2 terms to reach the final simplified form:

(1−x2)/((x2+1)2)

Final Answer

d()/d(x)x/(x2+1)=(1−x2)/((x2+1)2)

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