Find the Derivative - d/dx ((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.

ƒ(x)=(x2+7*x)/(x+1)

2. Identify the components for the quotient rule, where u=x2+7*x and v=x+1

u′=(d(x2)+7*x)/d(x)=2*x+7

v′=(d(x)+1)/d(x)=1

3. Apply the quotient rule formula d()/d(x)u/v=(u′*v−u*v′)/(v2)

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

4. Expand the terms in the numerator.

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

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

5. Subtract the expanded terms in the numerator.

2*x2+9*x+7−(x2+7*x)=2*x2+9*x+7−x2−7*x

x2+2*x+7

6. Combine the simplified numerator with the denominator.

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

Final Answer

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

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