Question

Find the one-sided derivatives of the function f(x) = |x+ 18|at the point x= -18 if they exist. If the derivative does not exist write DNE for your answer.

Answer 1

Left-hand derivative at x = -18 is -1

Right-hand derivative at x = -18 is 1

Step-by-step explanation:

Given the function;

`f(x)=|x+18|`

We'll use the below formula to find the left-hand derivative of the above function;

`f^(\prime)(a^-)=\lim _(h\to0^-)(f(a+h)-f(a))/(h)`

If we substitute and solve, we'll have;

`\begin{gathered} f^(\prime)(a^-)=\lim _(h\to0^-)=(|(-18+h)+18|-f(-18))/(h) \\ =\lim _(h\to0^-)=(|(-18+h)+18|-0)/(h) \\ =\lim _(h\to0^-)=(|h|)/(h) \end{gathered}`

Since this is a left-hand derivative, therefore h < 0;

`\lim _(h\to0^-)=(-h)/(h)=-1`

Let's go ahead and determine the right-hand derivative using the below formula;

`\begin{gathered} f^(\prime)(a^+)=\lim _(h\to0^+)(f(a+h)-f(a))/(h) \\ f^(\prime)(a^+)=\lim _(h\to0^+)=(|(-18+h)+18|-f(-18))/(h) \\ =\lim _(h\to0^+)=(|(-18+h)+18|-0)/(h) \\ =\lim _(h\to0^+)=(|h|)/(h) \end{gathered}`

Since this is a right-hand derivative, so h > 0;

`\lim _(h\to0^+)=(h)/(h)=1`

For a function to be differentiable at any point, its left-hand and right-hand derivative must exist and they must coincide.

From the above, we have that the left-hand derivative = -1 and the right-hand derivative = 1.