Science:Math Exam Resources/Courses/MATH110/December 2013/Question 01 (d)/Solution 1

False: If $f(x)$ is differentiable at $x=2$, then by the definition of derivative at a point, we have $\displaystyle f'(2)=\lim _{h\rightarrow 0}{\frac {f(2+h)-f(2)}{h}}.$ So to paraphrase the statement, we are asking whether $\displaystyle f'(2)=2$. Clearly, this does not have to be the case. Consider the example $\displaystyle f(x)=1$. We have $\displaystyle f(x)$ is differentiable at $x=2$ (and everywhere else too). But $f'(2)=0$.