What It Means to Be Differentiable
A function is differentiable at a point if it has a well-defined, finite derivative at that point. Geometrically, this means the graph has a unique, finite tangent slope there.
Most smooth graphs are differentiable at every point. But certain features of a graph prevent a derivative from existing. On the AP exam, the most important ones are corners, cusps, vertical tangents, and discontinuities.
The derivative at is defined by the limit of the difference quotient. If that limit does not exist as a finite number, then is not differentiable at .
The Four Main Reasons a Derivative Can Fail to Exist
- Discontinuity: The function is not continuous at the point.
- Corner: The left-hand and right-hand slopes are different.
- Cusp: The slopes become infinitely steep in opposite directions.
- Vertical tangent: The slope becomes infinite in the same direction.
The first fact to remember is extremely important:
In other words, differentiability implies continuity. But continuity does not necessarily imply differentiability.
1. Corners
A corner occurs when the graph changes direction abruptly and the slope approaching from the left is different from the slope approaching from the right.
The classic example is the absolute-value function:
The graph is continuous at , but it has a sharp point. To determine whether the derivative exists, compare the one-sided derivatives.
Determine whether is differentiable at .
This is exactly what the graph shows: the line approaches the origin with slope from the left and slope from the right.
2. Cusps
A cusp is a sharp point where the slope becomes infinitely steep, but the direction of the slope changes from one side to the other.
A standard AP Calculus example is:
At the origin, this graph has a cusp. Its derivative provides a quick way to see why.
Determine whether is differentiable at .
A cusp is therefore different from an ordinary corner. At a corner, the two slopes are finite but unequal. At a cusp, the slopes become unbounded in opposite directions.
3. Vertical Tangents
A vertical tangent occurs when the graph becomes infinitely steep and the slopes from both sides approach the same infinity.
Consider:
Its derivative is:
Determine whether is differentiable at .
Notice the important distinction: a vertical tangent is not the same thing as a cusp. For a vertical tangent, the slope approaches the same infinity from both sides. For a cusp, the slopes approach opposite infinities.
Corner vs. Cusp vs. Vertical Tangent
4. Discontinuities Also Prevent Differentiability
There is an even more fundamental reason a derivative can fail to exist: the function may not be continuous at the point.
If a function has a jump, hole, or vertical asymptote at , then it cannot have a derivative there.
This gives you an efficient strategy on the AP exam: check continuity first. If the function is discontinuous at the point, you can immediately conclude that it is not differentiable there.
5. Piecewise Functions
Piecewise-defined functions are especially important because the formula may change at a particular value. At that transition point, you need to check both continuity and the derivative.
Determine whether the following function is differentiable at :
How to Find All Points Where a Function Is Not Differentiable
When an AP Calculus problem asks you to identify all values where a function is not differentiable, don't simply differentiate the formula and look for places where the derivative "looks strange." Use a systematic process.
- Find points where the function is discontinuous. These automatically fail differentiability.
- Look for corners or cusps. Absolute values and piecewise definitions are common sources.
- Look for vertical tangents. These occur when the derivative becomes unbounded.
- Check endpoints. Depending on the definition being used, differentiability at an endpoint requires special attention because only one side of the domain is available.
- Check transition points in piecewise functions. Both continuity and matching one-sided derivatives matter.
A Useful AP Calculus Shortcut
If you are given a graph rather than an equation, you often don't need to calculate anything. Look at the shape of the graph.
A smooth curve with a well-defined finite tangent is usually differentiable. A sharp corner, cusp, vertical tangent, hole, jump, or asymptote is a warning sign that the derivative does not exist.
Common Mistakes
A function can be continuous but not differentiable. The graph of is continuous at the origin, but the corner means its derivative does not exist there.
A vertical tangent means the slope becomes unbounded. Since the derivative is required to be a finite real number, an infinite slope does not count as a derivative.
A corner has two different finite one-sided slopes. A cusp has slopes that become infinitely steep in opposite directions. The distinction matters when explaining why the derivative does not exist.
A derivative formula may be valid only on part of the domain. For piecewise functions, absolute values, and other special cases, inspect the original function and the behavior at the point itself.
Practice Problems
Determine whether is differentiable at .
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Determine whether is differentiable at .
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Determine whether is differentiable at .
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For what value of is the function continuous and differentiable?
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