Find the Derivative - d/dx y=((x+4)x)/(x+1)

Problem

d()/d(x)((x+4)*x)/(x+1)

Solution

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

y=(x2+4*x)/(x+1)

2. Identify the quotient rule formula, which states that for a function y=u/v the derivative is (vd(u)/d(x)−ud(v)/d(x))/(v2)

u=x2+4*x

v=x+1

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

d(x2+4*x)/d(x)=2*x+4

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

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

d(y)/d(x)=((x+1)*(2*x+4)−(x2+4*x)*(1))/((x+1)2)

5. Simplify the numerator by distributing the terms.

d(y)/d(x)=(2*x2+4*x+2*x+4−x2−4*x)/((x+1)2)

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

d(y)/d(x)=(x2+2*x+4)/((x+1)2)

Final Answer

d()/d(x)((x+4)*x)/(x+1)=(x2+2*x+4)/((x+1)2)

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