SOLVEDIs f(x) differentiable at x=1

cyt91

Given
$$\displaystyle f(x)=\left\{\begin{array}{cc}x^2,&\mbox{ if } x\leq 1\\\frac{x+1}{2}, & \mbox{ if } x>1\end{array}\right$$

Is f(x) differentiable at x=1?

I know that we have to prove
$$\displaystyle \boxed{ \lim_{\Delta x \to 0}\frac{f(1+\Delta x)-f(1)}{\Delta x}}$$

exist/does not exist at x=1. But how do I begin? It's a piece-wise function.

$$\displaystyle \lim \limits_{x \to a} \dfrac{f(x)-f(a)}{x - a}$$.
Then as $$\displaystyle x \to 1^{-}$$ and $$\displaystyle x \to 1^{+}$$ you can use the appropriate branch.