Find the Derivative - d/dx y=((x+7)x)/(x+1)

Problem

d()/d(x)((x+7)*x)/(x+1)

Solution

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

y=(x2+7*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+7*x

v=x+1

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

d(x2+7*x)/d(x)=2*x+7

d(x+1)/d(x)=1

4. Substitute these derivatives into the quotient rule formula.

d(y)/d(x)=((x+1)*(2*x+7)−(x2+7*x)*(1))/((x+1)2)

5. Expand the terms in the numerator.

(x+1)*(2*x+7)=2*x2+7*x+2*x+7=2*x2+9*x+7

(x2+7*x)*(1)=x2+7*x

6. Subtract the expanded terms to simplify the numerator.

2*x2+9*x+7−(x2+7*x)=x2+2*x+7

Final Answer

d()/d(x)((x+7)*x)/(x+1)=(x2+2*x+7)/((x+1)2)

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