Calculus > How to Figure Out When a Function is Not Differentiable
If there’s no limit to the slope of the secant line (in other words, if the limit does not exists at that point), then the derivative will not exist at that point. In general, a function is not differentiable in four separate scenarios: at a corner, at a cusp, at a vertical tangent, or at a jump discontinuity. You’ll be able to see these different types of scenarios by graphing the function on a graphing calculator; the only other way to “see” these events is algebraically — even if your algebra skills are very strong, it’s much easier and faster just to graph the function and look at the behavior.
How to Check for When a Function is Not Differentiable
Step 1: Check to see if the function has a distinct corner. For example, the graph of f(x)=|x – 1| has a corner at x=1, and is therefore not differentiable at that point:
Step 2: Look for a cusp in the graph. A cusp is slightly different from a corner — you can think of a cusp as a type of curved corner. This graph has a cusp at x = 0 (the origin):
Step 3: Look for a jump discontinuity. This normally happens in step or piecewise functions. The function may appear to not be continuous. the following graph jumps at the origin.
Step 4: Check for a vertical tangent. A vertical tangent is a line that runs straight up, parallel to the y-axis.
This graph has a vertical tangent at x=0.
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