Find the Derivative - d/dx ((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.

ƒ(x)=(x2+6*x)/(x+1)

2. Identify the quotient rule formula, which states that for a function u/v the derivative is (u′*v−u*v′)/(v2)

u=x2+6*x

v=x+1

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

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

d(v)/d(x)=1

4. Apply the quotient rule by substituting the functions and their derivatives into the formula.

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

5. Expand the terms in the numerator.

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

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

6. Subtract the expanded terms in the numerator and combine like terms.

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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