Find the Derivative - d/d@VAR f(x)=(2x+1)/(x+3)

Problem

d()/d(x)(2*x+1)/(x+3)

Solution

1. Identify the function as a quotient of two terms, u=2*x+1 and v=x+3 which requires the quotient rule.

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

3. Differentiate the numerator and denominator separately to find d(2*x+1)/d(x)=2 and d(x+3)/d(x)=1

4. Substitute these derivatives into the quotient rule formula.

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

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

d(ƒ(x))/d(x)=(2*x+6−2*x−1)/((x+3)2)

6. Combine the remaining constants in the numerator to reach the final simplified form.

d(ƒ(x))/d(x)=5/((x+3)2)

Final Answer

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

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