Proving differentiability comes down to showing that a function’s derivative exists at the point (or over the interval) in question. The core tool is the limit definition of the derivative: you compute the limit of the difference quotient and show it produces a finite value. If the limit exists, the function is differentiable there. If it doesn’t, the function isn’t.
This sounds simple in principle, but the technique changes depending on whether you’re working with a single-variable function at a point, checking an entire interval, or dealing with functions of multiple variables. Here’s how each case works.
The Difference Quotient Method
The most direct way to prove differentiability at a point x = a is to evaluate the limit that defines the derivative:
f'(a) = lim(x→a) [f(x) − f(a)] / (x − a)
If this limit exists and is finite, the function is differentiable at x = a. That’s the entire test. In practice, you substitute your function into this expression, simplify, and evaluate the limit. If you can reduce it to a concrete number, you’ve proven differentiability and found the derivative in one step.
An equivalent form uses h instead of x, writing the limit as h→0 of [f(a + h) − f(a)] / h. Both versions do the same thing. Pick whichever makes the algebra cleaner for the function you’re working with.
Checking Left and Right Limits
For piecewise functions or functions with potential trouble spots, you need to verify that the derivative approaches the same value from both sides. This means computing two one-sided limits separately:
- Left-hand derivative: lim(x→a⁻) [f(x) − f(a)] / (x − a)
- Right-hand derivative: lim(x→a⁺) [f(x) − f(a)] / (x − a)
If both one-sided limits exist and equal the same value, the function is differentiable at that point. If they disagree, it isn’t.
The classic example is f(x) = |x| at x = 0. Setting up the difference quotient gives lim(x→0) |x|/x. Approaching from the left, |x| equals −x (since x is negative), so the limit becomes lim(x→0⁻) (−x)/x = −1. Approaching from the right, |x| equals x, so the limit becomes lim(x→0⁺) x/x = 1. Since −1 ≠ 1, the limit doesn’t exist, and |x| is not differentiable at the origin. The graph has a sharp corner there, which is exactly the geometric intuition: you can’t draw a single tangent line at a corner.
Why Continuity Comes First
Every differentiable function is continuous. This means if you can show a function is discontinuous at a point, you’ve immediately proven it’s not differentiable there, no further work needed.
The reasoning is straightforward. If f is differentiable at a, then the derivative f'(a) exists as a finite limit. You can rewrite f(x) − f(a) as [(f(x) − f(a))/(x − a)] · (x − a). As x approaches a, the first factor approaches f'(a) (a finite number) and the second factor approaches 0. Their product approaches 0, which means f(x) approaches f(a). That’s exactly the definition of continuity.
The reverse is not true. A function can be continuous everywhere but still fail to be differentiable. The absolute value function is continuous at 0 (no jump, no gap) but not differentiable there. So continuity is necessary but not sufficient. Think of it as a quick screening test: check continuity first, and if the function fails, you can stop.
Where Differentiability Typically Fails
Functions fail to be differentiable at a point for a few recognizable reasons, and knowing these patterns saves time. You can often spot non-differentiability by inspection before doing any limit calculations.
- Corners and kinks: The function is continuous but changes direction abruptly, like |x| at x = 0. The left and right derivatives exist but disagree.
- Cusps: The function comes to a sharp point where the slope heads toward positive infinity from one side and negative infinity from the other. Think of f(x) = x^(2/3) at the origin.
- Vertical tangents: The slope becomes infinite. The function f(x) = x^(1/3) at x = 0 has a tangent line that’s perfectly vertical, so the derivative is undefined.
- Jumps and discontinuities: If the function isn’t continuous, it can’t be differentiable. A step function at its jump points is the simplest example.
If your function doesn’t fall into any of these categories at the point you’re checking, there’s a good chance it’s differentiable, but you still need to confirm with one of the methods above.
Proving Differentiability on an Interval
To prove a function is differentiable over an entire open interval (a, b), you need to show it’s differentiable at every point in that interval. For most standard functions built from polynomials, exponentials, trig functions, and their combinations through addition, multiplication, and composition, this follows from known differentiation rules. The sum, product, quotient, and chain rules all preserve differentiability wherever they apply.
For a closed interval [a, b], the requirements are slightly different. You need differentiability at every interior point in (a, b), plus the one-sided derivatives must exist at the endpoints: the right-hand derivative at x = a and the left-hand derivative at x = b.
In practice, proving differentiability on an interval often means identifying the trouble spots (points where the formula changes, denominators hit zero, or the function has a special form) and checking those individually. If the function is differentiable at every potential problem point and is built from differentiable pieces elsewhere, you’re done.
The Multivariable Case
For functions of two or more variables, differentiability is more subtle. A function f(x, y) is differentiable at a point if there exists a linear approximation that fits the function near that point with an error that shrinks faster than the distance to the point. Formally, there must be a linear map T such that the quantity [f(x) − f(a) − T(x − a)] / ‖x − a‖ goes to zero as x approaches a.
The matrix associated with this linear map is the Jacobian, the matrix of all partial derivatives. But here’s the critical subtlety: having all partial derivatives exist at a point does not guarantee differentiability. Partial derivatives only measure the rate of change along the coordinate axes. A function can behave nicely along each axis individually but badly along diagonal approaches.
The Continuous Partial Derivatives Shortcut
Directly verifying the limit definition in multiple variables is often tedious. The most widely used shortcut is the multidimensional differentiability theorem: if all partial derivatives of a function exist and are continuous in some neighborhood around a point, the function is differentiable at that point.
This gives a practical two-step method. First, compute all the partial derivatives. Second, check whether those partial derivatives are continuous functions. If both conditions hold in a region around your point, differentiability is guaranteed. For most functions you’ll encounter in a multivariable calculus course, this is the fastest path.
Differentiability Classes
You’ll sometimes see functions described as being “C¹” or “C².” These labels describe how smooth a function is. A function belongs to class C¹ on a region if its first derivative (or all its partial derivatives, in multiple dimensions) exists and is continuous throughout that region. A C² function has continuous second derivatives. More generally, Cᵏ means the function has continuous derivatives up through order k.
Being C¹ is a stronger condition than just being differentiable. A function can be differentiable everywhere but have a derivative that jumps around discontinuously. C¹ rules that out. If you can show a function is C¹ on an interval, you’ve proven not only that it’s differentiable but that its derivative behaves smoothly, which is often what’s needed in applications and further proofs.
Each class nests inside the next: every C² function is also C¹, and every C¹ function is differentiable and continuous. Checking the class of a function is straightforward. Differentiate it the required number of times and verify that the highest derivative you land on is continuous.