If $f(x)=\left\{\begin{array}{cc}\frac{x-1}{2 x^{2}-7 x+5} & ext { when } x
eq 1 \ -\frac{1}{3}

Question

If $f(x)=\left\{\begin{array}{cc}\frac{x-1}{2 x^{2}-7 x+5} & 	ext { when } x 
eq 1 \ -\frac{1}{3} & 	ext { when } x=1\end{array}ight.$ $f^{\prime}(1)$=

Professor PC Thomas Classes · Answer

We have\ $f^{\prime}(1)=\lim _{h ightarrow 0}\left[\frac{f(1+h)-f(1)}{h}ight]$\ $=\lim _{h ightarrow 0} \frac{\left[\frac{1+h-1}{2(1+h)^{2}-7(1+h)+5}-\left(-\frac{1}{3}ight)ight]}{h}$\ $=\lim _{h ightarrow 0}\left[\frac{3 h+2(1+h)^{2}-7(1+h)+5}{3 h\left[2(1+h)^{2}-7(1+h)+5ight]}ight]$\ $=\lim _{h ightarrow 0} \frac{2 h^{2}}{3 h\left(-3 h+2 h^{2}ight)}=\lim _{h ightarrow 0} \frac{2}{-9+6 h}=-\frac{2}{9}$\