Question

Find the derivatives from the left and from the right at \(x=1\) (if they exist). Is the function differentiable at \(x=1 ?\) \(f(x)=\left\\{\begin{array}{ll}x, & x \leq 1 \\ x^{2}, & x>1\end{array}\right.\)

Short answer

The function is not differentiable at \(x=1\) because the left-hand derivative (1) does not equal the right-hand derivative (2).

Step-by-step solution

Find the left-hand derivative

The left-hand derivative at \(x=1\) is the derivative of the function in the interval \(x\leq1\), which is \(f(x)=x\). The derivative of \(x\) with respect to \(x\) is simply 1. Therefore, the left-hand derivative at \(x=1\) is \(1\).

Find the right-hand derivative

The right-hand derivative at \(x=1\) is the derivative of the function in the interval \(x>1\), which is \(f(x)=x^{2}\). The derivative of \(x^{2}\) with respect to \(x\) is \(2x\). Therefore, the right-hand derivative at \(x=1\) is \(2\).

Check if the function is differentiable

A function is differentiable at a point if the left-hand derivative equals the right-hand derivative at that point. In this case, the left-hand derivative at \(x=1\) is 1, and the right-hand derivative at \(x=1\) is 2, therefore, they are unequal. Hence, the function is not differentiable at \(x=1\).