2. From the definition of the derivative (definition 5.3), find the derivative for each of the following functions: (a) f(x) = 6x (b) f(x) = 12x – 2 (c) f(x) = kx² for k a constant

Answer:a) [tex]f'(x)=6[/tex]b) [tex]f'(x)=12[/tex]c) [tex]f'(x)=2kx[/tex]Step-by-step explanation:To find :  From the definition of the derivative find the derivative for each of the following functions ?Solution : Definition of the derivative is [tex]f'(x)= \lim_{h \to 0}(\frac{f(x+h)-f(x)}{h})[/tex]Applying in the functions,a) [tex]f(x)=6x[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{6(x+h)-6x}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{6x+6h-6x}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{6h}{h})[/tex][tex]f'(x)=6[/tex]b) [tex]f(x)=12x-2[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{12(x+h)-2-(12x-2)}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{12x+12h-2-12x+2}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{12h}{h})[/tex][tex]f'(x)=12[/tex]c) [tex]f(x)=kx^2[/tex] for k a constant[tex]f'(x)= \lim_{h \to 0}(\frac{k(x+h)^2-kx^2}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{k(x^2+h^2+2xh-kx^2)}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{kx^2+kh^2+2kxh-kx^2}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(\frac{h(kh+2kx)}{h})[/tex][tex]f'(x)= \lim_{h \to 0}(kh+2kx)[/tex][tex]f'(x)=2kx[/tex]