Find the Derivative - d/dx g(x)=(x^2+5)/(x^2+6x)

Problem

d()/d(x)(x2+5)/(x2+6*x)

Solution

1. Identify the numerator u=x2+5 and the denominator v=x2+6*x to apply the quotient rule.

2. Differentiate the numerator and denominator separately.

d(u)/d(x)=2*x

d(v)/d(x)=2*x+6

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

d(g(x))/d(x)=((x2+6*x)*(2*x)−(x2+5)*(2*x+6))/((x2+6*x)2)

4. Expand the terms in the numerator.

(x2+6*x)*(2*x)=2*x3+12*x2

(x2+5)*(2*x+6)=2*x3+6*x2+10*x+30

5. Subtract the expanded terms and simplify the numerator.

(2*x3+12*x2)−(2*x3+6*x2+10*x+30)

2*x3−2*x3+12*x2−6*x2−10*x−30

6*x2−10*x−30

6. Factor the denominator if necessary, though leaving it as a square is standard.

(x2+6*x)2=(x*(x+6))2=x2*(x+6)2

Final Answer

d()/d(x)(x2+5)/(x2+6*x)=(6*x2−10*x−30)/(x2*(x+6)2)

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