Difficulty: Intermediate | Prerequisites: Limits (Section 2.1), continuity, slope of a tangent line concept
A function is differentiable at a point when you can draw a single, well-defined tangent line there. Corners, cusps, vertical tangents and discontinuities all break differentiability. Reading a graph for these features, then stating the differentiable intervals in interval notation, is a core exam skill in MTH 132 Section 2.2.
Differentiable (at a point)
A function f is differentiable at x = a when the limit that defines f'(a) exists and is finite. Graphically, the curve has a single, non-vertical tangent line at that point.
In simple terms, the graph is "smooth" there, with no sharp turns, jumps or vertical spikes.
Differentiable (on an interval)
f is differentiable on an open interval (a, b) when it is differentiable at every point in that interval.
Think of it as: the entire stretch of curve between a and b is smooth and unbroken.
Corner (or sharp turn)
A point where the graph changes direction abruptly, so the slope from the left and the slope from the right are different finite values. The one-sided derivatives both exist but disagree.
In simple terms, the curve makes a "V" shape (or an inverted "V").
Cusp
A point where both one-sided derivatives are infinite but with opposite signs, creating a sharp pointed tip. For instance, the graph of f(x) = x^(2/3) at the origin.
Think of it as a needle-sharp point where the tangent line would be vertical from both sides but pointing in opposite directions.
Vertical tangent
A point where the tangent line exists but is vertical, meaning the derivative tends to positive or negative infinity. The function is continuous there but not differentiable.
In simple terms, the curve goes straight up or straight down at that instant.
Discontinuity
Any point where the function is not continuous: a jump, a hole, or an asymptote. Differentiability requires continuity, so every discontinuity automatically kills differentiability at that point.
Think of it as: if the graph is broken, you cannot draw a tangent line across the break.
The homework (Problem 1) gives a graph and asks you to state, in interval notation, where f(x) is differentiable. The method:
Start by noting the full domain of the function from the graph.
Remove every x-value where you spot a corner, cusp, vertical tangent, discontinuity (jump, hole, or asymptote), or an endpoint of the domain.
Write the remaining x-values as a union of open intervals.
For example, if a piecewise graph is smooth on (-2, 1), has a corner at x = 1, and is smooth again on (1, 4), the answer is (-2, 1) ∪ (1, 4). Endpoints of the domain are excluded because differentiability requires a two-sided limit.
Key visual cues to scan for:
A sharp change of direction (corner): the graph bends suddenly, like the tip of |x|.
An open or closed circle on the graph: indicates a possible discontinuity. Check whether the function value matches the limit from both sides.
A segment that goes vertical: the tangent line would be vertical, so the derivative is undefined.
A gap or jump in the curve: a discontinuity, which rules out differentiability.
Problem 2 asks you to sketch f'(x) from the graph of f(x), then match it to one of eight choices. The procedure:
Where f is increasing, f' is positive (above the x-axis).
Where f is decreasing, f' is negative (below the x-axis).
Where f has a local maximum or minimum (a hilltop or valley), f' = 0 (the derivative graph crosses or touches the x-axis).
Where f is linear (a straight segment), f' is constant (a horizontal segment at whatever that slope is).
Where f is concave up (curving like a bowl), f' is increasing.
Where f is concave down (curving like a dome), f' is decreasing.
At any point where f is not differentiable (corner, cusp, vertical tangent, discontinuity), the derivative graph has no value; mark it with an open circle or simply leave a gap.
Work left to right across the graph of f. Estimate the slope at several points and plot those slope values on a new set of axes. Connect them smoothly wherever the original function is smooth.
Differentiability implies continuity: if f is differentiable at x = a, then f is necessarily continuous there. The converse is false: f(x) = |x| is continuous at x = 0 but not differentiable there (it has a corner). This one-way implication is tested constantly.
Students often think that continuity is enough for differentiability. It is not. A function can be perfectly continuous yet have a corner or cusp where no single tangent line exists.
Students sometimes exclude closed-circle endpoints from the differentiable interval. If the function is smooth on both sides of a closed dot (and it is an interior point), it is differentiable there. The dot itself is about the function value, not the derivative.
When sketching the derivative, students frequently confuse the height of f with the slope of f. The derivative graph plots the slope at each point, not the y-value of the original function. A function can be at its highest point and have a derivative of zero.
Students occasionally forget that a vertical tangent means the derivative does not exist at that point. The tangent line does exist (it is vertical), but the slope is undefined, so differentiability fails.
⚠️ "State where f(x) is differentiable" is a direct exam question. You must give your answer in interval notation using unions, with parentheses (not brackets) around every interval.
⚠️ Matching a derivative graph to the original function graph is a standard multiple-choice item. Practise going left to right, reading the slope at each region, and plotting it.
⚠️ The statement "differentiable implies continuous" (and its converse being false) appears on nearly every MTH 132 exam in some form. Know a counterexample: f(x) = |x| at x = 0.
True or false: If f is differentiable at x = 3, then f is continuous at x = 3.
True or false: If f is continuous at x = 3, then f is differentiable at x = 3.
Fill in the blank: At a local maximum of f(x), the value of f'(x) is ______.
True or false: Where f(x) is concave up, f'(x) is increasing.
Fill in the blank: A corner on the graph of f means that the left-hand and right-hand ______ are not equal.
Answers: 1. True. 2. False (counterexample: |x| at 0). 3. Zero. 4. True. 5. Derivatives (or slopes).
Q: A piecewise function f is continuous everywhere on [-3, 5]. It has corners at x = -1 and x = 2, and a vertical tangent at x = 4. Where is f differentiable?
A: (-3, -1) ∪ (-1, 2) ∪ (2, 4) ∪ (4, 5). Corners and vertical tangents break differentiability at those points; endpoints of the domain are excluded.
Q: Suppose f(x) is increasing and concave down on the interval (1, 4). Describe the derivative f'(x) on that interval.
A: f'(x) is positive (because f is increasing) and decreasing (because f is concave down) on (1, 4).
Q: The graph of f has a horizontal tangent at x = 2. What can you conclude about f'(2)?
A: f'(2) = 0, because the slope of the tangent line is zero.
Q: Give an example of a function that is continuous at a point but not differentiable there. Name the point and the reason.
A: f(x) = |x| at x = 0. The function is continuous, but it has a corner (the left-hand slope is -1 and the right-hand slope is +1), so it is not differentiable.
Q: On a graph of f(x), you see a segment that is a straight line with negative slope from x = 3 to x = 5. What does f'(x) look like on that interval?
A: f'(x) is a negative constant on (3, 5), because a linear segment has constant slope.
This material connects directly to the limit definition of the derivative (Section 2.2, Problems 3 to 9), which is the algebraic machinery behind everything here. The graphical intuition in these notes tells you what to expect; the limit definition tells you how to compute it.
It also connects to continuity (Section 2.1): since differentiability implies continuity, every theorem about continuous functions (Intermediate Value Theorem, for instance) automatically applies wherever a function is differentiable.
Later in the course, the relationship between f and f' you practised here by sketching will become central to curve sketching and optimisation (Sections 4.1 to 4.3).
Related Terms / Search Tags
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