Find Where Increasing/Decreasing Using Derivatives f(x)=x/(x^2+9)

Problem

ƒ(x)=x/(x^2+9)

Solution

1. Find the derivative of the function using the quotient rule, where u=x and v=x^2+9

ƒ(x)^′=((x^2+9)d(x)/d(x)−xd(x^2+9)/d(x))/((x^2+9)^2)

2. Simplify the derivative by calculating the individual derivatives and combining like terms.

ƒ(x)^′=((x^2+9)*(1)−x*(2*x))/((x^2+9)^2)

ƒ(x)^′=(x^2+9−2*x^2)/((x^2+9)^2)

ƒ(x)^′=(9−x^2)/((x^2+9)^2)

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

(9−x^2)/((x^2+9)^2)=0

9−x^2=0

x^2=9

x=−3,3

4. Test intervals around the critical points to determine the sign of ƒ(x)^′ The denominator (x^2+9)^2 is always positive, so the sign depends only on the numerator 9−x^2
   For (−∞,−3) test x=−4 9−(−4)^2=−7 (Negative).
   For (−3,3) test x=0 9−(0)^2=9 (Positive).
   For (3,∞) test x=4 9−(4)^2=−7 (Negative).

5. Determine behavior based on the signs. The function increases where ƒ(x)^′>0 and decreases where ƒ(x)^′<0

Final Answer

Increasing: *(−3,3), Decreasing: *(−∞,−3)∪(3,∞)

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