Find the Derivative - d/dx y=((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.

y=(x2+9*x)/(x+1)

2. Identify the components for the quotient rule, where u=x2+9*x and v=x+1

d(y)/d(x)=(vd(u)/d(x)−ud(v)/d(x))/(v2)

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 formula by substituting the derivatives and original functions.

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

5. Distribute the terms in the numerator to prepare for simplification.

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

6. Combine like terms in the numerator to reach the final simplified form.

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

Final Answer

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

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