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

Problem

d()/d(x)*((x+6)⋅x/(x+1))

Solution

1. Simplify the expression before differentiating by multiplying the terms to form a single rational function.

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

2. Identify the numerator and denominator for the quotient rule, where u=x2+6*x and v=x+1

u=x2+6*x

v=x+1

3. Differentiate u and v with respect to x

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

d(v)/d(x)=1

4. Apply the quotient rule formula, which is d()/d(x)u/v=(v⋅d(u)/d(x)−u⋅d(v)/d(x))/(v2)

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

5. Expand the terms in the numerator.

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

6. Combine like terms in the numerator to find the final derivative.

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

Final Answer

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

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