Find Where Increasing/Decreasing Using Derivatives f(x)=x/(x^2+1)

Problem

ƒ(x)=x/(x2+1)

Solution

1. Find the derivative of the function using the quotient rule, where u=x and v=x2+1

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

2. Simplify the numerator to find the expression for the first derivative.

d(ƒ(x))/d(x)=(x2+1−2*x2)/((x2+1)2)

d(ƒ(x))/d(x)=(1−x2)/((x2+1)2)

3. Identify critical points by setting the derivative equal to zero and solving for x

(1−x2)/((x2+1)2)=0

1−x2=0

x=1,x=−1

4. Determine the sign of the derivative in the intervals created by the critical points: (−∞,−1) (−1,1) and (1,∞)

For *x=−2,ƒ′*(−2)=(1−(−2)2)/(((−2)2+1)2)=(−3)/25<0

For *x=0,ƒ(0)′=(1−0)/((0+1)2)=1>0

For *x=2,ƒ(2)′=(1−2)/((2+1)2)=(−3)/25<0

5. Conclude the intervals based on the sign of the derivative. The function is increasing where ƒ(x)′>0 and decreasing where ƒ(x)′<0

Final Answer

Increasing: *(−1,1), Decreasing: *(−∞,−1)∪(1,∞)

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