Question

When is |x+1| diffrentiable

Answer 1

yo all values of x except x=-1

Answer 2

How do you know?

Answer 3

it's 1

Answer 4

the derivative of x is 1 and the derivative of 1 is 0, add them together, you would get 1

Answer 5

a function is not differentiable when its graph is having a pin point

Answer 6

More formally, a function is not differentiable at \(x=c\) if
\[\lim_{x\to c^-}f'(x)\not=\lim_{x\to c^+}f'(x)\]
In this case, since \(f(x)=|x+1|=\begin{cases}x+1&\text{for }x>1\\0&\text{for }x=1\\-(x+1)&\text{for }x<1\end{cases}\), you have
\[f'(x)=\begin{cases}1&\text{for }x>1\\DNE&\text{for }x=1&&\text{(because the limit-condition is not met)}\\-1&\text{for }x<1\end{cases}\]