1. **State the problem:** Find the derivative of the function $f(x) = \frac{1}{x^2+1}$ using first principles (the definition of the derivative). 2. **Recall the definition of the derivative:** $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. **Apply the function to the definition:** $$f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)^2+1} - \frac{1}{x^2+1}}{h}$$ 4. **Find a common denominator for the difference in the numerator:** $$= \lim_{h \to 0} \frac{\frac{x^2+1 - ((x+h)^2+1)}{((x+h)^2+1)(x^2+1)}}{h}$$ 5. **Simplify the numerator inside the fraction:** $$(x^2+1) - ((x+h)^2+1) = x^2 + 1 - (x^2 + 2xh + h^2 + 1) = -2xh - h^2$$ 6. **Substitute back:** $$f'(x) = \lim_{h \to 0} \frac{\frac{-2xh - h^2}{((x+h)^2+1)(x^2+1)}}{h} = \lim_{h \to 0} \frac{-2xh - h^2}{h \cdot ((x+h)^2+1)(x^2+1)}$$ 7. **Factor out $h$ in the numerator:** $$= \lim_{h \to 0} \frac{h(-2x - h)}{h \cdot ((x+h)^2+1)(x^2+1)}$$ 8. **Cancel $h$ from numerator and denominator:** $$= \lim_{h \to 0} \frac{-2x - h}{((x+h)^2+1)(x^2+1)}$$ 9. **Take the limit as $h \to 0$:** $$f'(x) = \frac{-2x}{(x^2+1)^2}$$ **Final answer:** $$\boxed{f'(x) = \frac{-2x}{(x^2+1)^2}}$$