How To Prove Differentiability At A Point Or Interval?

Differentiability is a property of a function at a specific point or across an entire interval. To prove differentiability at a point, you show that the limit definition of the derivative exists and is finite at that point. To prove differentiability on an interval, you show the function is differentiable at every point in that interval. That is the core of it. The rest is about how you actually do that in practice.

What Does It Actually Mean for a Function to Be Differentiable?

A function is differentiable at a point if the derivative exists at that point. The derivative is defined as the limit of the difference quotient as the change in x approaches zero. If that limit exists and produces a real number, the function is differentiable there.

If the limit does not exist, or if it goes to infinity, the function is not differentiable at that point. That is not a technicality. It has real geometric meaning. A function that is not differentiable at a point has no well-defined tangent line there.

There are three common ways differentiability fails at a point:

• A sharp corner or cusp, where the left-hand and right-hand limits of the difference quotient disagree
• A vertical tangent, where the limit is infinite rather than a finite number
• A discontinuity, where the function is not even continuous at that point

Differentiability implies continuity. If a function is differentiable at a point, it must be continuous there. The reverse is not true. A function can be continuous everywhere and still fail to be differentiable at one or more points. The absolute value function is the standard example. It is continuous at zero but has a sharp corner there, and the derivative does not exist at that point.

How Do You Prove Differentiability at a Single Point?

You prove differentiability at a point by working directly with the limit definition of the derivative. There is no shortcut that replaces this step when you need a rigorous proof.

The formal approach has three parts:

1. Write out the difference quotient: the quantity f(x) minus f(a), all divided by x minus a
2. Take the limit of that quotient as x approaches a
3. Show that the limit exists and equals a finite real number

If the limit exists and is finite, the function is differentiable at that point. If the limit fails to exist, or is infinite, it is not.

For functions built from polynomials, sines, cosines, and exponentials, you can often verify differentiability at a point by confirming the function is defined there, is continuous there, and that its derivative formula produces a finite value at that point. This works because these functions are known to be differentiable on their entire domains. But if you are asked to prove it from scratch, you need the limit computation.

One common mistake is to assume that because a function looks smooth, it must be differentiable. Smoothness of a graph is not a proof. You need either a direct limit computation or a theorem that guarantees differentiability.

How Do You Prove Differentiability on an Interval?

To prove differentiability on an open interval, you show the function is differentiable at every point in that interval. In practice, this usually means applying known theorems rather than checking every point one at a time.

The standard tools are:

• Polynomials are differentiable everywhere
• Sine and cosine are differentiable everywhere
• Exponential functions are differentiable everywhere
• Sums, differences, products, quotients, and compositions of differentiable functions are differentiable wherever they are defined

So if you can express a function as a combination of these building blocks, you can conclude it is differentiable on any interval where all the pieces are defined and the operations are valid.

For a closed interval, the situation is slightly different. Differentiability on a closed interval [a, b] typically means the function is differentiable on the open interval (a, b) and has a one-sided derivative at each endpoint. The one-sided derivative at a is the limit as x approaches a from the right. At b, it is the limit as x approaches b from the left.

This distinction matters for theorems like the Mean Value Theorem, which requires continuity on the closed interval and differentiability on the open interval.

What Is the Relationship Between Continuity and Differentiability?

Differentiability is the stronger condition. Every function that is differentiable at a point is also continuous at that point. But not every continuous function is differentiable.

This is one of the most commonly misunderstood points in calculus. Students often assume that continuity is enough for differentiability. It is not.

The Weierstrass function is the classic counterexample on a larger scale. It is continuous everywhere but differentiable nowhere. Functions like this show that continuity and differentiability are genuinely different properties, not just different words for the same thing.

For a proof, the implication goes one direction only. If you know a function is differentiable at a point, you can conclude it is continuous there. If you know it is continuous, you cannot conclude anything about differentiability. You have to check separately.

What Are Common Mistakes When Proving Differentiability?

The most common mistake is confusing continuity with differentiability. A function can be continuous at a point and still fail to have a derivative there. Checking continuity is not enough.

Another mistake is assuming that if the derivative formula gives a finite value, the function must be differentiable. This is usually true for elementary functions, but it skips the underlying reasoning. If the function is defined piecewise, or if there is any question about the domain, you need to check the limit definition directly at the point in question.

A third mistake is ignoring endpoints when working on a closed interval. Differentiability on [a, b] requires one-sided derivatives at a and b. If you only check the interior, your proof is incomplete.

Finally, some students try to prove differentiability by graphing the function and observing that it looks smooth. A graph can suggest differentiability, but it cannot prove it. Proof requires the limit computation or a valid theorem.

When Can You Use Theorems Instead of the Limit Definition?

You can use theorems whenever the function is built from known differentiable pieces using operations that preserve differentiability. This covers most functions you encounter in a standard calculus course.

The key theorems are:

• The sum rule: if f and g are differentiable, so is f + g
• The product rule: if f and g are differentiable, so is fg
• The quotient rule: if f and g are differentiable and g is not zero, so is f/g
• The chain rule: if f and g are differentiable, so is their composition

These rules let you conclude differentiability without returning to the limit definition every time. But they only apply where the conditions are met. If a denominator is zero, or if a composition involves a function that is not differentiable at some point, the theorem does not apply there.

For piecewise functions, you usually need to check the boundary points individually. The function may be differentiable on each piece but fail at the point where the pieces meet. You check this by computing the left-hand and right-hand limits of the difference quotient at that point.

How Does Differentiability Relate to Tangent Lines and Linear Approximation?

Differentiability at a point means the function has a well-defined tangent line at that point. The slope of that tangent line is the derivative. This is not just a geometric convenience. It is the foundation of linear approximation.

When a function is differentiable at a point, you can approximate its values near that point using the tangent line. The approximation gets better the closer you are to the point. This is the basis for many numerical methods and for the way derivatives are used in physics, engineering, and economics.

If a function is not differentiable at a point, there is no tangent line, and linear approximation breaks down there. The function may still be continuous, but it does not behave like a straight line at small scales near that point.

This is why differentiability matters beyond the classroom. It tells you when a function can be treated as locally linear, which is the assumption behind most applications of calculus.