124. By computing the limit $\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$, find the derivatives of the following functions:
1) $y=3 x^{2}-4 x$
2) $y=\frac{1}{x}$
3) $y=\sqrt{x}$
4) $y=\cos 3 x$.
Solution: Guided by the specified general rule for directly finding the derivative, we sequentially find:

1) For the function $y=3 x^{2}-4 x$:
I) $y+\Delta y=3(x+\Delta x)^{2}-4(x+\Delta x)=3 x^{2}+6 x \Delta x+$

$$
+3 \Delta x^{2}-4 x-4 \Delta x
$$

II) $\Delta y=\left(3 x^{2}+6 x \Delta x+3 \Delta x^{2}-4 x-4 \Delta x\right)-\left(3 x^{2}-4 x\right)=$

$$
=6 x \Delta x+3 \Delta x^{2}-4 \Delta x
$$

III) $\frac{\Delta y}{\Delta x}=\frac{6 x \Delta x+3 \Delta x^{2}-4 \Delta x}{\Delta x}=6 x+3 \Delta x-4$;

IV) $\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim (6 x+3 \Delta x-4)=6 x-4$.

Therefore, $\frac{d y}{d x}=\frac{d\left(3 x^{2}-4 x\right)}{d x}=6 x-4$.

2) For the function $y=\frac{1}{x}$:
I) $y+\Delta y=\frac{1}{x+\Delta x}$

II) $\Delta y=\frac{1}{x+\Delta x}-\frac{1}{x}=-\frac{\Delta x}{x(x+\Delta x)}$;

III) $\frac{\Delta y}{\Delta x}=-\frac{1}{x(x+\Delta x)}$;

IV) $\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim \left[-\frac{1}{x(x+\Delta x)}\right]=-\frac{1}{x^{2}}$.

Therefore, $\left(\frac{1}{x}\right)^{\prime}=-\frac{1}{x^{2}}$.

3) For the function $y=\sqrt{x}$:
I) $y+\Delta y=\sqrt{x+\Delta x}$

II) $\Delta y=\sqrt{x+\Delta x}-\sqrt{x}$

III) $\frac{\Delta y}{\Delta x}=\frac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}$;

IV) $y^{\prime}=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim \frac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}=$

$=\lim \frac{x+\Delta x-x}{\Delta x(\sqrt{x+\Delta x}+\sqrt{x})}=\frac{1}{2 \sqrt{x}} \cdot$

4) For the function $y=\cos 3 x$:
I) $y+\Delta y=\cos 3(x+\Delta x)$

II) $\Delta y=\cos 3(x+\Delta x)-\cos 3 x=$

$$
=-2 \sin \left(3 x+\frac{3}{2} \Delta x\right) \sin \frac{3}{2} \Delta x
$$

III) $\frac{\Delta y}{\Delta x}=-\frac{2 \sin \left(-\frac{3}{2} \Delta x\right) \sin \frac{3}{2} \Delta x}{\Delta x}$;

IV) $y^{\prime}=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=-2 \lim \sin \left(3 x+\frac{3}{2} \Delta x\right) \cdot \lim \frac{\sin \frac{3}{2} \Delta x}{\Delta x}=$ $=-2 \sin 3 x \cdot \operatorname{tim} \frac{\frac{3}{2} \Delta x}{\Delta x}=-2 \sin 3 x \cdot \frac{3}{2}=-3 \sin 3 x . *$