2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Most functions that you will see in this course are differentiable, which means that you can take their derivative. The exam, however, will likely throw in a few functions whose derivative does not exist. Here are the rules to make sure a function is differentiable at a certain point. 

First, the function must be continuous at that point. It makes sense that if a point doesn’t exist on a function, you can’t determine its instantaneous rate of change. ➡️➡️➡️

Second, if you calculate the derivative from the left, it should equal the derivative when calculated from the right. This is applicable to absolute value functions, which change directions suddenly. Since absolute value graphs are made up of two functions, the place where those two functions meet does not have a derivative. ➡️ = ⬅️

Example: Determine the range of values on f(x)= x+1 where f'(x) exists.

f(x) = { x + 1   (-∞, -1)

x - 1   (-1, ∞)

lim x->-1(-) f'(x) = lim x->-1(-) (-1) = -1

lim x->-1(+) f'(x) = lim x->-1(+) (1) = 1

(since the derivative is the slope of the tangent line, the derivative of a line will simply be the slope of that line)  lim x->-1(-) f'(x) =/= lim x->-1(+) f'(x) Therefore, the derivative doesn’t exist at x = -1