Respuesta :
Given:
[tex]f(x)=4x^2+3x-10.[/tex]The formula to find the derivative of the function by using limit is
[tex]f^{\prime}(x)=\lim _{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]Replace x=x+h in the given function f(x) to find f(x+h).
[tex]f(x+h_{})=4(x+h)^2+3(x+h)-10.[/tex][tex]f(x+h_{})=4(x^2+2hx+h^2)+3x+3h-10.[/tex][tex]f(x+h_{})=4x^2+8hx+4h^2+3x+3h-10.[/tex]Substitute know values in the given formula, we get
[tex]f^{\prime}(x)=\lim _{h\to0}\frac{(4x^2+8hx+4h^2+3x+3h-10)-(4x^2+3x-10)}{h}[/tex][tex]f^{\prime}(x)=\lim _{h\to0}\frac{4x^2+8hx+4h^2+3x+3h-10-4x^2-3x-10}{h}[/tex][tex]f^{\prime}(x)=\lim _{h\to0}\frac{4x^2-4x^2+8hx+4h^2+3x-3x+3h-10-10}{h}[/tex][tex]f^{\prime}(x)=\lim _{h\to0}\frac{8hx+4h^2+3h}{h}[/tex]Taking out the common term h, we get
[tex]f^{\prime}(x)=\lim _{h\to0}\frac{h(8x+4h+3)}{h}[/tex][tex]f^{\prime}(x)=\lim _{h\to0}(8x+4h+3)[/tex]Taking limit, we get
[tex]f^{\prime}(x)=8x+4(0)+3[/tex][tex]f^{\prime}(x)=8x+3[/tex]Hence the derivative of the given function is 8x+3.
