Ҳосилаи функсияи зерин ёфта шавад:
\[y = 4x^5-7+x-\frac{1}{x^3}+\sqrt{x}+e^x.\]

Ҳал.

\(\begin{multline}
y' = \left(4x^5-7+x-\frac{1}{x^3}+\sqrt{x}+e^x \right)' = \\
= \left(4x^5\right)'-\left(7\right)'+\left(x\right)'-\left(\frac{1}{x^3}\right)'+\left(\sqrt{x}\right)'+\left(e^x \right)';
\end{multline}\)

\((4x^5)'=4\cdot 5\cdot x^{5-1}=20x^4;\)
\((7)'=0;\)
\((x)'=1;\)
\((\frac{1}{x^3})'=-3\cdot x^{-3-1}=-3\cdot x^{-4}=-\frac{3}{x^4};\)
\((\sqrt{x})'=\frac{1}{2}\cdot x^{\frac{1}{2}-1}=\frac{1}{2}\cdot x^{-\frac{1}{2}}=\frac{1}{2\cdot\sqrt{x}};\)
\((e^x)'=e^x.\)

Пас
\(y'=20x^4-0+1-(-\frac{3}{x^4})+\frac{1}{2\sqrt{x}}+e^x\)
\(y'=20x^4+1+\frac{3}{x^4}+\frac{1}{2\sqrt{x}}+e^x\)

Ҷавоб.

\[y'=20x^4+1+\frac{3}{x^4}+\frac{1}{2\sqrt{x}}+e^x.\]