Question

Which one of the following graphs represents the derivative \( f'(x) = \frac{df}{dx} \) of the function \( f(x) = \frac{1}{1+x^2} \) most closely (graphs are schematic and not drawn to scale)? 

Hint:

When taking the derivative of rational functions like \( \frac{1}{1+x^2} \), use the quotient rule or recognize it as a standard derivative.

Answer 1

Step 1: Finding the derivative.
The function \( f(x) = \frac{1}{1+x^2} \) is a rational function, and its derivative is given by applying the quotient rule or recognizing it as a standard function. The derivative is \( f'(x) = -\frac{2x}{(1+x^2)^2} \), which has the shape of a curve that is symmetric about the origin and decreases as \( |x| \) increases.
Step 2: Conclusion.
Thus, the correct answer is option (B), as the graph of \( f'(x) \) matches the expected shape of the derivative.