Find the Derivative - d/dx ((x+9)x)/(x+1)

Problem

d()/d(x)((x+9)*x)/(x+1)

Solution

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

(x+9)*x=x2+9*x

2. Identify the quotient rule formula for a function u/v which is (vd(u)/d(x)−ud(v)/d(x))/(v2)

u=x2+9*x

v=x+1

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

(d(x2)+9*x)/d(x)=2*x+9

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

4. Apply the quotient rule by substituting the components into the formula.

d()/d(x)(x2+9*x)/(x+1)=((x+1)*(2*x+9)−(x2+9*x)*(1))/((x+1)2)

5. Simplify the numerator by distributing and combining like terms.

(x+1)*(2*x+9)=2*x2+9*x+2*x+9=2*x2+11*x+9

(2*x2+11*x+9)−(x2+9*x)=x2+2*x+9

Final Answer

d()/d(x)((x+9)*x)/(x+1)=(x2+2*x+9)/((x+1)2)

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