What is Differentiability of a Function?

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Grade Level:

Class 12

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

Definition

What is it?

Differentiability of a function means whether you can find a unique tangent line (a line that just touches the curve at one point) at every point on its graph. If a function is differentiable at a point, its graph is smooth and continuous, without any sharp corners, breaks, or jumps at that point.

Simple Example

Quick Example

Imagine you're driving an auto-rickshaw on a road. If the road is smooth and has gentle curves, your steering wheel turns smoothly. This is like a differentiable function. If you suddenly hit a sharp turn or a speed breaker, your steering has to jerk, which is like a point where a function is NOT differentiable.

Worked Example

Step-by-Step

Let's check if the function f(x) = |x| is differentiable at x = 0.

Step 1: Understand the definition of differentiability. A function is differentiable at a point 'c' if its derivative exists at 'c'. This means the left-hand derivative (LHD) and right-hand derivative (RHD) must be equal at 'c'.

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Step 2: Calculate the Left-Hand Derivative (LHD) at x = 0. The formula for LHD is lim (h->0-) [f(c+h) - f(c)] / h. Here, c = 0, and f(x) = |x|. So, f(0) = |0| = 0. For h < 0, |0+h| = |h| = -h.
LHD = lim (h->0-) [(-h) - 0] / h = lim (h->0-) [-h / h] = lim (h->0-) [-1] = -1.

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Step 3: Calculate the Right-Hand Derivative (RHD) at x = 0. The formula for RHD is lim (h->0+) [f(c+h) - f(c)] / h. Here, c = 0, and f(x) = |x|. For h > 0, |0+h| = |h| = h.
RHD = lim (h->0+) [(h) - 0] / h = lim (h->0+) [h / h] = lim (h->0+) [1] = 1.

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Step 4: Compare LHD and RHD. We found LHD = -1 and RHD = 1.

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Step 5: Since LHD is not equal to RHD (-1 ≠ 1), the derivative of f(x) = |x| does not exist at x = 0.

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Answer: The function f(x) = |x| is NOT differentiable at x = 0 because it has a sharp corner (a 'kink') at that point.

Why It Matters

Understanding differentiability helps engineers design smooth curves for car bodies or airplane wings, ensuring less air resistance. In AI/ML, it's crucial for training models efficiently, like how Google Maps finds the shortest route. Doctors use it to model how medicines spread in the body, optimizing dosages for patients.

Common Mistakes

MISTAKE: Thinking that if a function is continuous, it must also be differentiable. | CORRECTION: While differentiability implies continuity, the reverse is not true. A function can be continuous but not differentiable, like f(x) = |x| at x=0 (it's continuous but has a sharp corner).

MISTAKE: Assuming a function is differentiable at points where its graph has a vertical tangent. | CORRECTION: A vertical tangent means the slope is infinite, which means the derivative is undefined. So, a function is not differentiable at such points.

MISTAKE: Not checking both Left-Hand Derivative (LHD) and Right-Hand Derivative (RHD) when dealing with piecewise functions or functions involving absolute values. | CORRECTION: For differentiability at a point, LHD and RHD must both exist and be equal. Always calculate both to confirm.

Practice Questions

Try It Yourself

QUESTION: Is the function f(x) = x^2 differentiable at x = 1? | ANSWER: Yes

QUESTION: At what point(s) is the function f(x) = |x - 2| not differentiable? | ANSWER: x = 2

QUESTION: Consider the function f(x) = { x + 1 for x <= 0, x^2 + 1 for x > 0 }. Is f(x) differentiable at x = 0? | ANSWER: Yes

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's graph must have a sharp corner at that point.

The function must be undefined at that point.

The Correct Answer Is:

For a function to be differentiable at a point, it must first be continuous at that point. A sharp corner or being undefined at a point would mean it's not differentiable.

Real World Connection

In the Real World

Imagine the speed of a train as it leaves a station. If the train accelerates smoothly, its speed function is differentiable. If the driver suddenly applies brakes or accelerates very sharply (which doesn't happen in real trains for safety!), that would be a point of non-differentiability. Engineers use differentiability to ensure smooth transitions in speed and acceleration for comfortable rides and efficient fuel use, even in our Indian Railways.

Key Vocabulary

Key Terms

TANGENT LINE: A straight line that touches a curve at a single point without crossing it | DERIVATIVE: The rate at which a function's value changes at a given point; it represents the slope of the tangent line | CONTINUITY: A function is continuous if its graph can be drawn without lifting the pen from the paper (no breaks or jumps) | LEFT-HAND DERIVATIVE (LHD): The derivative of a function approached from the left side of a point | RIGHT-HAND DERIVATIVE (RHD): The derivative of a function approached from the right side of a point

What's Next

What to Learn Next

Now that you understand differentiability, you're ready to dive into 'Rules of Differentiation'. This will teach you how to easily find derivatives for many common functions without having to use the long limit definition every time. Keep going, you're doing great!