Examining differentiability Using Graph of Function

Condition for Differentiability

Differentiability describes whether a function has a smooth tangent or slope at a given point. In simple terms, a function $f(x)$ is differentiable at a point if it has a unique, well-defined derivative there. If not, the function is said to be non-differentiable.

This section explains the mathematical condition for differentiability, the definition of derivative, and the criteria using left-hand and right-hand limits.

Definition of Differentiability

Let $f$ be a real-valued function and $c$ be a point in its domain. The derivative of $f$ at $c$ is defined as

$>\lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$

provided this limit exists and is finite.

The derivative of $f$ at $c$ is denoted by $>f'(c)$ or $>\left.\frac{d}{dx}f(x)\right|_{x=c}$.

For all $x$ in the domain of $f$, the function that gives the derivative is written as

$>f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

The process of finding this derivative is called differentiation. When we say differentiate $f(x)$ with respect to $x$, we mean to find $f'(x)$.

Left-Hand and Right-Hand Derivatives

A function $f(x)$ is said to be differentiable at $x = c$ if both the left-hand derivative (LHD) and right-hand derivative (RHD) exist and are equal.

That is,

$>\lim_{h \to 0^-} \frac{f(c + h) - f(c)}{h} = \lim_{h \to 0^+} \frac{f(c + h) - f(c)}{h}$

and both limits are finite.

If either limit does not exist or if they are not equal, the function is not differentiable at that point.

Condition for Differentiability

A function $f(x)$ is differentiable at $x = c$ if:

• $f(x)$ is continuous at $x = c$, and

• Both $>R_f'(c)$ and $>L_f'(c)$ exist and are equal.

If these conditions fail, $f(x)$ is non-differentiable at $x = c$.

Why Differentiability Matters

Differentiability ensures the function behaves smoothly, without any sharp turns, cusps, or breaks. Functions that are differentiable everywhere have continuous slopes and are fundamental in studying motion, optimization, and curve analysis in calculus.

Examining Differentiability Using Differentiation and Graph

To determine whether a function is differentiable at a particular point, we can use two key approaches - algebraic differentiation (based on limits) and graphical analysis (based on the shape of the curve).

This section explains both methods in detail and highlights common cases where functions fail to be differentiable, such as sharp corners, discontinuities, or vertical tangents.

1. Using Differentiation (Applicable Only for Continuous Functions)

For a function to be differentiable at a certain point, it must first be continuous there. If the function is not continuous, it cannot be differentiable.

In many cases, functions are piecewise-defined, meaning they are expressed differently in different intervals. To test differentiability for such functions, follow these steps:

Step 1: Check Continuity

If a function is defined as:

$>f(x) =
\begin{cases}
g_1(x), & x < a \\
g_2(x), & x \ge a
\end{cases}$

First, verify if $f(x)$ is continuous at $x = a$.
If it is not continuous, differentiability automatically fails.

Step 2: Differentiate Each Part

If the function is continuous, differentiate both sides:

$>f'(x) =
\begin{cases}
(g_1(x))', & x < a \\
(g_2(x))', & x > a
\end{cases}$

Step 3: Compare Left and Right Derivatives

Now compare the left-hand derivative (LHD) and right-hand derivative (RHD) at $x = a$:

$>\lim_{x \to a^-} (g_1'(x))$ and $>\lim_{x \to a^+} (g_2'(x))$

If these two limits exist and are equal, $f(x)$ is differentiable at $x = a$.
If they are unequal or do not exist, the function is not differentiable at that point.

2. Using Graphical Analysis

A quicker way to judge differentiability is by examining the graph of the function. Visually, differentiability corresponds to the graph being smooth — no corners, breaks, or vertical tangents.

A function $f(x)$ is not differentiable at $x = a$ if:

1. The function is discontinuous at $x = a$.

   • Any break or gap in the graph means the derivative cannot exist.

2. The graph has a sharp turn or cusp at $x = a$.

   • The slope suddenly changes direction, making $f'(x)$ undefined.

3. The function has a vertical tangent at $x = a$.

   • The slope tends to infinity, implying an undefined derivative.

Check the differentiability of the following function.
1. f(x)=sin⁡|x|

Illustration 1

Method 1

Using graphical transformation, we can draw its graph

When you plot the graph of $f(x) = \sin|x|$, it appears smooth on both sides of the y-axis but forms a sharp corner at $x = 0$.

This sharp turn indicates that the slope of the tangent changes abruptly at that point. Since differentiability requires a continuous and smooth change in slope, we can immediately conclude that $f(x)$ is not differentiable at $x = 0$.

Graphically, the function remains continuous, but the derivative does not exist at the corner point.

Method 2

As $LHL = RHL = f(0) = 0$, so the function is continuous at $x = 0$.
So we can use differentiation to check differentiability.

$f(x) =
\begin{cases}
-\sin x, & x < 0 \\
\sin x, & x \ge 0
\end{cases}$

$\therefore f'(x) =
\begin{cases}
-\cos x, & x < 0 \\
\cos x, & x > 0
\end{cases}$

$\therefore LHD = f'(0^-) = -1$ and $RHD = f'(0^+) = 1$

As these are not equal, so $f(x) = \sin|x|$ is not differentiable at $x = 0$.

Illustration 2

$f(x) = |\log|x||$, $x \ne 0$

Plot the graph of $|\log|x||$ using graphical transformation.

We can see that graph has a sharp turn at +1 and -1 so the function is not differentiable at these points.

Solved Examples Based On Condition of Differentiability:

Example 1: Let $f,g:\mathbb{R} \rightarrow \mathbb{R}$ be two functions defined by
$f(x)=
\begin{cases}
x\sin\left(\dfrac{1}{x}\right), & x \ne 0 \\
0, & x=0
\end{cases}$
and $g(x)=x f(x)$  [JEE Main 2014]

Statement I: $f$ is a continuous function at $x=0$.
Statement II: $g$ is a differentiable function at $x=0$.

1. Both statements I and II are false.

2. Both statements I and II are true.

3. Statement I is true, Statement II is false.

4. Statement I is false, Statement II is true.

Solution

As we learned in the Condition for Differentiability
A function $f(x)$ is said to be differentiable at $x = x_0$ if $Rf'(x_0)$ and $Lf'(x_0)$ both exist and are equal; otherwise, it is non-differentiable.

$f(x)=
\begin{cases}
x\sin\left(\dfrac{1}{x}\right), & x \ne 0 \\
0, & x = 0
\end{cases}$
and $g(x)=x f(x)$

$\lim_{x \to 0^+/0^-} x\sin\dfrac{1}{x} = 0 \times \text{finite} = 0$

So $f(x)$ is continuous at $x=0$.

Now,
$g(x)=
\begin{cases}
x^2 \sin\dfrac{1}{x}, & x \ne 0 \\
0, & x=0
\end{cases}$

$\lim_{h \to 0^+} \dfrac{h^2 \sin\dfrac{1}{h} - 0}{h} = h\sin\dfrac{1}{h} = 0$
$\lim_{h \to 0^-} \dfrac{h^2 \sin\dfrac{1}{h} - 0}{h} = h\sin\dfrac{1}{h} = 0$

$\therefore g'(0^+) = g'(0^-)$.

So, $g(x)$ is differentiable at $x=0$.

Example 2: Let $f(x)=
\begin{cases}
-1, & -2 \le x < 0 \\
x^2 - 1, & 0 \le x \le 2
\end{cases}$
and $g(x)=|f(x)|+f(|x|)$.

Then, in the interval $(-2,2)$, $g$ is:  [JEE Main 2019]

1. not differentiable at two points

2. differentiable at all points

3. not differentiable at one point

4. not continuous

Solution

Condition for differentiability:
A function $f(x)$ is said to be differentiable at $x = x_0$ if $Rf'(x_0)$ and $Lf'(x_0)$ both exist and are equal; otherwise, it is non-differentiable.

Properties of differentiable functions:
At every corner point, $f(x)$ is continuous but not differentiable.
For example, $|x - a|$ is continuous but not differentiable at $x = a$, for $a > 0$.

Here, $y = f(x)$ has only one non-differentiable point at $x = 1$.

Example 3: Let $f:(-1,1) \to \mathbb{R}$ be a function defined by
$f(x)=\max{-|x|, -1 - x^2}$.

If $K$ is the set of all points at which $f$ is not differentiable, then $K$ has exactly:  [JEE Main 2019]

1. two elements

2. three elements

3. five elements

4. one element

Solution

Properties of differentiable functions:
At every corner point, $f(x)$ is continuous but not differentiable.
For example, $|x - a|$ is continuous but not differentiable at $x = a$, for $a > 0$.

Condition for differentiability:
A function $f(x)$ is said to be differentiable at $x = x_0$ if $Rf'(x_0)$ and $Lf'(x_0)$ both exist and are equal; otherwise, it is non-differentiable.

Here, $f:(-1,1) \to \mathbb{R}$ and
$f(x)=\max{-|x|, 1 - x^2}$

Hence, $f(x)$ is non-differentiable at 3 points in $(-1,1)$.

Example 4: Let $f:[0,3] \to \mathbb{R}$ be defined by
$f(x)=\min{x - [x], 1 + [x] - x}$ where $[x]$ is the greatest integer less than or equal to $x$.

Let $P$ denote the set containing all $x \in [0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x \in (0,3)$ where $f$ is not differentiable.

Then the sum of the number of elements in $P$ and $Q$ is equal to:  [JEE Main 2021]

1. 5

2. 4

3. 2

4. 1

Solution

$f(x)=\min({x}, 1 - {x})$

No point of discontinuity ⇒ $n(P)=0$
5 points of non-differentiability ⇒ $n(Q)=5$

$\Rightarrow n(P)+n(Q)=5$

Example 5: Let $f:[0,\infty) \to [0,3]$ be a function defined by
$f(x)=
\begin{cases}
\max{\sin t : 0 \le t \le x}, & 0 \le x \le \dfrac{\pi}{2} \\
1 + \cos x, & x > \pi
\end{cases}$

Which of the following is true?  [JEE Main 2021]

1. $f$ is continuous everywhere but not differentiable exactly at one point in $(0,\infty)$

2. $f$ is differentiable everywhere in $(0,\infty)$

3. $f$ is not continuous exactly at two points in $(0,\infty)$

4. $f$ is continuous everywhere but not differentiable exactly at two points in $(0,\infty)$

Solution

Is differentiable everywhere in $(0,\infty)$.
The option (2) is correct.

List of topics related to Examining differentiability Using Graph of Function

This section provides a detailed list of all key subtopics linked to differentiability, graphical interpretation, and related calculus concepts - helping you identify what areas to revise and practice thoroughly.

Differentiability and Existence of Derivative

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NCERT Resources

This section compiles all NCERT-based resources for Chapter 5 – Continuity and Differentiability, including structured notes, solved textbook questions, and exemplar exercises for complete exam preparation.

NCERT Class 12 Maths Notes for Chapter 5 - Continuity and Differentiability

NCERT Class 12 Maths Solutions for Chapter 5 - Continuity and Differentiability

NCERT Class 12 Maths Exemplar Solutions for Chapter 5 - Continuity and Differentiability

Practice Questions based on Examining differentiability Using Graph of Function

Here you’ll find topic-wise practice questions and MCQs designed to test your understanding of graphical differentiability, continuity, and related theorems effectively.

Examining Differentiability Using Graph Of Function- Practice Question MCQ

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