Given
[tex]f(x)=4x^2[/tex]Find
derivative of the function
Explanation
By limit definition ,
[tex]\frac{dy}{dx}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]so, we have
[tex]\begin{gathered} \frac{dy}{dx}=\lim_{h\to0}\frac{\lbrace4\left(x+h\right)^2-4x^2\rbrace}{h} \\ \frac{dy}{dx}=\lim_{h\to0}\frac{\lbrace4(x^2+h^2+2xh)^-4x^2\rbrace}{h} \\ \frac{dy}{dx}=\lim_{h\to0}\frac{\lbrace4x^2+4h^2+8xh-4x^2\rbrace}{h} \\ \frac{dy}{dx}=\lim_{h\to0}\frac{8xh+4h^2}{h} \\ \frac{dy}{dx}=\lim_{h\to0}8x+4h \\ \frac{dy}{dx}=8x \end{gathered}[/tex]Final Answer
The derivative of the function is 8x
