170. Differentiate the function $y=x^{2}$.
Solution. $1^{0} . y_{\text {n }}=(x+\Delta x)^{2}=x^{2}+2 x \Delta x+(\Delta x)^{2}$.

$2^{0} . \Delta y=y_{\mathrm{H}}-y=\left(x^{2}+2 x \Delta x+(\Delta x)^{2}\right)-x^{2}=2 x \Delta x+(\Delta x)^{2}$.

$3^{0} \cdot \frac{\Delta y}{\Delta x}=\frac{2 x \Delta x+(\Delta x)^{2}}{\Delta x}=\frac{2 x \Delta x}{\Delta x}+\frac{(\Delta x)^{2}}{\Delta x}=2 x+\Delta x$.

4. $\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim _{\Delta x \rightarrow 0}(2 x+\Delta x)=2 x$.

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