1. Muammo: $\arctan\left(\frac{x}{y}\right)$ ifodasining $y$ bo'yicha hosilasini toping. 2. Formulalar va qoidalar: Agar $f(y) = \arctan(u(y))$ bo'lsa, hosila quyidagicha hisoblanadi: $$f'(y) = \frac{u'(y)}{1 + u(y)^2}$$ Bu yerda $u(y) = \frac{x}{y}$. 3. $u(y)$ ning hosilasini topamiz: $$u(y) = \frac{x}{y} = x \cdot y^{-1}$$ $$u'(y) = x \cdot (-1) y^{-2} = -\frac{x}{y^2}$$ 4. Endi hosilani ifodalaymiz: $$f'(y) = \frac{-\frac{x}{y^2}}{1 + \left(\frac{x}{y}\right)^2} = \frac{-\frac{x}{y^2}}{1 + \frac{x^2}{y^2}}$$ 5. Quyidagi qadamda kasrlarni birlashtiramiz: $$f'(y) = \frac{-\frac{x}{y^2}}{\frac{y^2 + x^2}{y^2}} = -\frac{x}{y^2} \cdot \frac{y^2}{y^2 + x^2}$$ 6. $y^2$ larni qisqartiramiz: $$f'(y) = -\frac{x}{\cancel{y^2}} \cdot \frac{\cancel{y^2}}{y^2 + x^2} = -\frac{x}{y^2 + x^2}$$ Natija: $$\frac{d}{dy} \arctan\left(\frac{x}{y}\right) = -\frac{x}{y^2 + x^2}$$