[tex]1)\; y=x^2\\\\y`(x)= \lim_{\Delta x \to \ 0} \frac{\Delta y}{\Delta x} \\\\f(x)=x^2\\\\\Delta y=f(x+\Delta x)-f(x)\\\\\Delta y=(x+\Delta x)^2-x^2=x^2+2x\Delta x+(\Delta x)^2-x^2=2x\Delta x+(\Delta x)^2[/tex]

[tex]\lim_{\Delta x \to 0} \frac{2x\Delta x+(\Delta x)^2}{\Delta x}= \lim_{\Delta x \to 0}(2x+\Delta x)=2x+0=2x[/tex]

[tex]y`(x)=(x^2)`=2x[/tex]

[tex]2)\; y=4x^2\\\\y`(x)= \lim_{\Delta x \to \ 0} \frac{\Delta y}{\Delta x} \\\\f(x)=4x^2\\\\\Delta y=f(x+\Delta x)-f(x)\\\\\Delta y=4(x+\Delta x)^2-4x^2=4x^2+8x\Delta x+4(\Delta x)^2-4x^2=8x\Delta x+4(\Delta x)^2[/tex]

[tex]\lim_{\Delta x \to 0} \frac{8x\Delta x+4(\Delta x)^2}{\Delta x}= \lim_{\Delta x \to 0}(8x+\Delta x)=8x+0=8x\\\\y`(x)=(4x^2)=8x[/tex]