Respuesta :
Answer:
[tex]f'(x)=\dfrac{37}{(1-6x)^2}[/tex]
Domain of f(x): [tex](-\infty,\frac{1}{6})\cup (\frac{1}{6},\infty)[/tex].
Domain of f'(x): [tex]Domain=(-\infty,\frac{1}{6})\cup (\frac{1}{6},\infty)[/tex].
Step-by-step explanation:
Consider the given function
[tex]f(x)=\dfrac{6+x}{1-6x}[/tex]
Domain of this function is all real numbers except those numbers for which denominator is equal to 0.
[tex]1-6x=0[/tex]
[tex]1=6x[/tex]
Divide both sides by 6.
[tex]\frac{1}{6}=x[/tex]
The function f(x) is not defined for 1/6. So, the domain of the function f(x) is
[tex]Domain=(-\infty,\frac{1}{6})\cup (\frac{1}{6},\infty)[/tex]
Find the derivative of the function using the definition of derivative.
[tex]f'(x)=lim_{x\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}[/tex]
[tex]f'(x)=lim_{x\rightarrow 0}\dfrac{\dfrac{6+(x+h)}{1-6(x+h)}-\dfrac{6+x}{1-6x}}{h}[/tex]
[tex]f'(x)=lim_{x\rightarrow 0}\frac{1}{h}\dfrac{(6+x+h)(1-6x)-(6+x)(1-6x-6h)}{(1-6x-6h)(1-6x)}[/tex]
[tex]f'(x)=lim_{x\rightarrow 0}\frac{1}{h}\dfrac{(6+x)(1-6x)+h(1-6x)-(6+x)(1-6x)-(6+x)(-6h)}{(1-6x-6h)(1-6x)}[/tex]
[tex]f'(x)=lim_{x\rightarrow 0}\frac{h}{h}\dfrac{1-6x+36+6x}{(1-6x-6h)(1-6x)}[/tex]
[tex]f'(x)=lim_{x\rightarrow 0}\dfrac{37}{(1-6x-6h)(1-6x)}[/tex]
Apply limit.
[tex]f'(x)=\dfrac{37}{(1-6x-6(0))(1-6x)}[/tex]
[tex]f'(x)=\dfrac{37}{(1-6x)^2}[/tex]
The derivative of given function is [tex]f'(x)=\dfrac{37}{(1-6x)^2}[/tex].
[tex](1-6x)^2=0[/tex]
[tex]1-6x=0[/tex]
[tex]1=6x[/tex]
Divide both sides by 6.
[tex]\frac{1}{6}=x[/tex]
The function f'(x) is not defined for 1/6. So, the domain of the function f'(x) is
[tex]Domain=(-\infty,\frac{1}{6})\cup (\frac{1}{6},\infty)[/tex].
