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 $f^{\prime }(2)=\underset{h\to 0}{\lim }\frac{f(2+h)-f(2)}{h}.$ So to paraphrase the statement, we are asking whether $f^{\prime }(2)=2$. Clearly, this does not have to be the case. Consider the example $f(x)=1$. We have $f(x)$ is differentiable at $x=2$ (and everywhere else too). But $f^{\prime }(2)=0$.