Ҳосилаи функсияи зерин ёфта шавад:
\[f(x) = \frac{2x+5}{3x^5+4x-9}.\]

Ҳал. Барои ҳалли ин мисол аз формулаи зерин истифода мебарем:
\[\left(\frac{g_1(x)}{g_2(x)}\right)'=\frac{g_2(x)g_1'(x)-g_1(x)g_2'(x)}{[g(x)]^2}.\]
\(\begin{multline}
f'(x)=\left(\frac{2x+5}{3x^5+4x-9}\right)' = \frac{(3x^5+4x-9)\cdot (2x+5)'-(2x+5)\cdot (3x^5+4x-9)'}{(3x^5+4x-9)^2} = \\
= \frac{(3x^5+4x-9)\cdot 2-(2x+5)\cdot (3\cdot 5\cdot x^4+4)}{(3x^5+4x-9)^2} = \\
= \frac{6x^5+8x-18-30x^5-8x-75x^4-20}{(3x^5+4x-9)^2}= \\
= \frac{-24x^5-75x^4-38}{(3x^5+4x-9)^2} = -\frac{24x^5+75x^4+38}{(3x^5+4x-9)^2}.
\end{multline}\)

Ҷавоб.

\[f'(x) = -\frac{24x^5+75x^4+38}{(3x^5+4x-9)^2}.\]