What is Differentiability at a Point?

S7-SA1-0022

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Differentiability at a point means that a function has a well-defined, unique tangent line at that specific point. Imagine drawing a perfectly smooth curve; if you can place a ruler (tangent line) at any point without it 'wobbling' or having a sharp corner, then the function is differentiable there. It essentially means the function is 'smooth' and has no sharp turns or breaks at that point.

Simple Example

Quick Example

Think about the journey of an auto-rickshaw. If the auto is moving smoothly on a straight road, its speed at any exact moment is clear and steady. But if it suddenly takes a very sharp, jerky turn or stops abruptly, its speed at that exact moment of the sharp change becomes harder to define smoothly. Differentiability is like that smooth, predictable speed at a point.

Worked Example

Step-by-Step

Let's check if the function f(x) = x^2 is differentiable at x = 1.

Step 1: Understand differentiability. A function is differentiable at a point 'a' if the limit of [f(x) - f(a)] / [x - a] exists as x approaches 'a'. This is called the derivative.
---Step 2: Find f(a) for a = 1. So, f(1) = 1^2 = 1.
---Step 3: Set up the limit expression. We need to find the limit as x approaches 1 of [f(x) - f(1)] / [x - 1], which is [x^2 - 1] / [x - 1].
---Step 4: Simplify the expression. We know that x^2 - 1 can be factored as (x - 1)(x + 1). So the expression becomes (x - 1)(x + 1) / (x - 1).
---Step 5: Cancel out (x - 1) from the numerator and denominator (since x is approaching 1 but not equal to 1). This leaves us with (x + 1).
---Step 6: Evaluate the limit. As x approaches 1, (x + 1) approaches (1 + 1) = 2.
---Step 7: Since the limit exists and is a finite number (2), the function f(x) = x^2 is differentiable at x = 1.

Answer: The derivative of f(x) = x^2 at x = 1 is 2, so it is differentiable at x = 1.

Why It Matters

Understanding differentiability is super important for many cool fields! In AI/ML, it helps algorithms learn by finding the 'best path' for improvement. In physics, it's used to calculate instantaneous speed or acceleration of rockets or cars. Engineers use it to design smooth curves for bridges or roller coasters, ensuring safety and efficiency.

Common Mistakes

MISTAKE: Thinking that if a function is continuous, it must be differentiable. | CORRECTION: Continuity is necessary for differentiability, but not sufficient. A function can be continuous but have a sharp 'corner' (like the absolute value function at x=0), making it non-differentiable there.

MISTAKE: Trying to find the derivative at a point where the function is undefined or has a 'jump'. | CORRECTION: A function must be continuous at a point to even consider differentiability. If there's a break or a hole, it cannot be differentiable.

MISTAKE: Confusing differentiability with just finding the derivative formula. | CORRECTION: Differentiability at a point specifically asks if the derivative *exists* at that single point, meaning the left-hand derivative and right-hand derivative must be equal and finite.

Practice Questions

Try It Yourself

QUESTION: If a function f(x) has a vertical tangent line at a point, is it differentiable at that point? | ANSWER: No, if a function has a vertical tangent line at a point, its derivative is undefined (approaches infinity), so it is not differentiable at that point.

QUESTION: Consider the function f(x) = x if x >= 0 and f(x) = -x if x < 0. Is this function differentiable at x = 0? Explain why. | ANSWER: No. This function is f(x) = |x|. The left-hand derivative at x=0 is -1, and the right-hand derivative at x=0 is 1. Since they are not equal, the function is not differentiable at x = 0.

MCQ

Quick Quiz

Which of the following conditions is necessary for a function to be differentiable at a point?

The function must be constant at that point.

The function must be continuous at that point.

The function must be zero at that point.

The function must be a straight line at that point.

The Correct Answer Is:

For a function to be differentiable at a point, it must first be continuous there. If there's a break or jump, a unique tangent line cannot be drawn. Options A, C, and D are not general requirements.

Real World Connection

In the Real World

Imagine a drone delivering a package for a company like Zepto. Its flight path needs to be smooth and predictable. If the path has sharp, sudden changes (points of non-differentiability), the drone might crash or the package could be damaged. Engineers use differentiability to design smooth trajectories, ensuring safe and efficient delivery.

Key Vocabulary

Key Terms

CONTINUITY: A function is continuous if its graph can be drawn without lifting the pen. | TANGENT LINE: A straight line that touches a curve at exactly one point without crossing it at that point. | DERIVATIVE: The rate at which a function changes at a specific point, representing the slope of the tangent line. | SMOOTH CURVE: A curve without any sharp corners or breaks.

What's Next

What to Learn Next

Great job understanding differentiability at a point! Next, you should explore 'Rules of Differentiation'. This will teach you quick methods to find derivatives for different types of functions, which builds directly on knowing when a function can be differentiated.