Question

The derivative of $f\left(x\right)=\{\begin{array}{cc}\frac{x-1}{2x^2-7x+5}& whenx\ne 1\\ -1/3& whenx=1\end{array}atx=1.$ is $=-\frac{2}{a}$ Find a

Answer 1

$f\left(x\right)=\frac{x-1}{(x-1)(2x-5)}=\frac{1}{2x-5},x\ne 1$
$f\left(x\right)=-1/3,x=1$
$R{f}^{\prime }\left(1\right)=\underset{h\to 0}{lim}\frac{f(1+h)-f\left(1\right)}{h}=\underset{h\to 0}{lim}\frac{\frac{1}{2h-3}+\frac{1}{3}}{h}$ $=\underset{h\to 0}{lim}\frac{2h}{3h(2h-3)}=-\frac{2}{9}$
Similarly $L{f}^{\prime }\left(1\right)=-2/9$
$\therefore f^{\prime }\left(1\right)=-2/9$
Ans: 9