Find the Derivative - d/dx y=((x+6)x)/(x+1)

Problem

d()/d(x)((x+6)*x)/(x+1)

Solution

1. Expand the numerator to simplify the expression before differentiating.

y=(x2+6*x)/(x+1)

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

u=x2+6*x

v=x+1

3. Differentiate the numerator and the denominator with respect to x

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

d(v)/d(x)=1

4. Substitute these derivatives into the quotient rule formula.

d(y)/d(x)=((x+1)*(2*x+6)−(x2+6*x)*(1))/((x+1)2)

5. Expand the terms in the numerator.

(x+1)*(2*x+6)=2*x2+6*x+2*x+6=2*x2+8*x+6

(x2+6*x)*(1)=x2+6*x

6. Subtract the expanded terms to simplify the numerator.

2*x2+8*x+6−(x2+6*x)=x2+2*x+6

Final Answer

d()/d(x)((x+6)*x)/(x+1)=(x2+2*x+6)/((x+1)2)

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