Examining differentiability Using Graph of Function

Edited By Komal Miglani | Updated on Jul 02, 2025 07:51 PM IST

Differentiability is an important concept in calculus. It is useful in understanding the rate of a change in the function. The existence of a derivative at a point implies that the function has a specific rate of change at that point. These concepts of differentiability have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

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Condition for differentiability
Examining Differentiability Using Differentiation and Graph
Solved Examples Based On Condition of Differentiability:

Examining differentiability Using Graph of Function

In this article, we will cover the concept of the Condition of differentiability. This topic falls under the broader category of Calculus, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE. A total of nine questions have been asked on this topic in JEE Main from 2013 to 2023, one question in 2014, two questions in 2019, four in 2021, and two in 2023.

Condition for differentiability

A function $\mathrm{f}(\mathrm{x})$ is said to be differentiable if $R f^{\prime}\left(x_{\circ}\right)$ and $L f^{\prime}\left(x_{\circ}\right)$ both exist and are equal otherwise nondifferentiable

Suppose f is a real function and c is a point in its domain. The derivative of $f$ at c is defined by

$
\lim\limits_{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}
$

provided this limit exists.
Derivative of f at c is denoted by $\mathrm{f}^{\prime}$ (c) or $\left.\frac{d}{d x}(f(x))\right|_c$
The function defined by

$
f^{\prime}(x)=\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}
$
The process of finding the derivative of a function is called differentiation. We also use the phrase differentiate $f(x)$ concerning $x$ to mean find $f$ '(x)Whenever we defined derivative, we had put a caution provided the limit exists. Now the natural question is; what if it doesn't? The question is quite pertinent and so is its answer. If $\lim\limits_{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$

it does not exist, we say that $f$ is not differentiable at $c$. In other words, we say that a function f is differentiable at a point c in its domain if both $\lim\limits_{h \rightarrow 0^{-}} \frac{f(c+h)-f(c)}{h} \lim\limits_{h \rightarrow 0^{+}} \frac{f(c+h)-f(c)}{h}$ are finite and equal.

Examining Differentiability Using Differentiation and Graph

1. Using Differentiation (only for continuous functions)

Some functions are defined piecewise, in such cases first we need to check if the function is continuous at the split point, and if it is continuous we need to differentiate each branch function and compare left-hand and right-hand derivative at the split point.

$
f(x)= \begin{cases}g_1(x), & x<a \\ g_2(x), & x \geq a\end{cases}
$
First check if $f(x)$ is continuous at $x=a$. If it is not continuous, then it cannot be differentiable. If it is continuous, then to check differentiability, find

$
f^{\prime}(x)= \begin{cases}\left(g_1(x)\right)^{\prime}, & x<a \\ \left(g_2(x)\right)^{\prime}, & x>a\end{cases}
$
Differentiability can be checked at $\mathrm{x}=\mathrm{a}$ by comparing

$
\lim\limits_{x \rightarrow a^{-}}\left(g_1(x)\right)^{\prime} \text { and } \lim\limits_{x \rightarrow a^{+}}\left(g_2(x)\right)^{\prime}
$

2. Differentiability using Graphs

A function $f(x)$ is not differentiable at $x=a$ if
1. The function is discontinuous at $x=a$
2. The graph of a function has a sharp turn at $x=a$
3. A function has a vertical tangent at $x=a$

Illustration 1

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

Method 1

Using graphical transformation, we can draw its graph

Using the graph we can tell that at $x=0$, the graph has a sharp turn, so it is not differentiable at $x=0$.

Method 2

As $L H L=R H L=f(0)=0$, so the function is continuous at $x=0$
So we can use differentiation to check differentiability

$
\begin{aligned}
& \quad \mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}
-\sin x, & x<0 \\
\sin x, & x \geq 0
\end{array}\right. \\
& \therefore \quad \quad \mathrm{f}^{\prime}(\mathrm{x})=\left\{\begin{array}{cc}
-\cos x, & x<0 \\
\cos x, & x>0
\end{array}\right. \\
& \therefore \quad \text { LHD }=\mathrm{f}^{\prime}\left(0^{-}\right)=-1 \text { and } \mathrm{RHD}=\mathrm{f}^{\prime}\left(0^{+}\right)=1
\end{aligned}
$
As these are not equal, so, $f(x)=\sin |x|$ is not differentiable at $x=0$

Illustration 2

$
f(x)=\|\log \mid x\|, x \text { not equal to } 0
$
Plot the graph of | log $|\mathrm{x}|$ | 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 $f, g: R \rightarrow R$ Be two functions defined by $f(x)=\left\{\begin{array}{cl}x \sin \left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x=0\end{array}\right.$ and $g(x)=x f(x)$: [JEE Main 2014]

Statement I :$f$ is a continuous function at $\mathrm{x}=0$.
Statement II : g is a differentiable function at $\mathrm{x}=0$.
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
Condition for differentiable -
A function $\mathrm{f}(\mathrm{x})$ is said to be differentiable at $x=x_0$ if $R f^{\prime}\left(x_{\circ}\right)$ and $L f^{\prime}\left(x_{\circ}\right)$ both exist and are equal otherwise non differentiable

$
\begin{aligned}
& f(x)=\left\{\begin{array}{cc}
x \sin \left(\frac{1}{x}\right) & x \neq 0 \\
0 & x=0
\end{array}\right. \\
& \text { and } g(x)=x f(x) \\
& \lim\limits_{x \rightarrow 0^{+} / 0^{-}} \quad x \sin \frac{1}{x}=0 \times \text { finite }=0
\end{aligned}
$
So $f(x)$ is continuous at $x=0$

$
\begin{aligned}
& g(x)=\left\{\begin{array}{cl}
x^2 \sin \frac{1}{x} & x \neq 0 \\
0 & x=0
\end{array}\right. \\
& \lim\limits_{h \rightarrow 0^{+}} \frac{h^2 \sin \frac{1}{h}-0}{h}=h \sin \frac{1}{h}=0 \\
& \lim\limits_{h \rightarrow 0^{-}} \frac{-h^2 \sin \frac{1}{h}-0}{-h}=h \sin \frac{1}{h}=0 \\
& \therefore g^{\prime}\left(0^{+}\right)=g^{\prime}\left(0^{-}\right)
\end{aligned}
$

So, $\mathrm{g}(\mathrm{x})$ is differentiable at $x=0$

Example 2: Let $f(x)=\left\{\begin{array}{cc}-1, & -2 \leq x<0 \\ x^2-1, & 0 \leq x \leq 2\end{array}\right.$ and $g(x)=|f(x)|+f(|x|)$.Then, in the interval $(-2,2), \mathrm{g}$ 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 $\mathrm{f}(\mathrm{x})$ is said to be differentiable at $x=x_{\circ}$ if $R f^{\prime}\left(x_{\circ}\right)$ and $L f^{\prime}\left(x_{\circ}\right)$ both exist and are equal otherwise non differentiable

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

$
y=f(x)
$

only one nondifferential point at $x=1$

Example 3: Let $f:(-1,1) \rightarrow R_{\text {be a function defined by }}$ $f(x)=\max \left\{-|x|,-\sqrt{1-x^2}\right\}$. If K be 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.
ex: $|x-a|$ is continuous but not differentiable at $x=a$ for $a>0$
- wherein

$
\begin{aligned}
& f:(-1,1) \rightarrow R \\
& f(x)=\max \left\{-|x|, \sqrt{1-x^2}\right\}
\end{aligned}
$
Non-differentiable at 3 points in $(-1,1)$.

Example 4: Let $f:[0,3] \rightarrow \mathbf{R}_{\text {be defined by }}$ $f(x)=\min \{x-[x], 1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal to x . Let P denote the set containing all $\mathrm{x} \in[0,3]_{\text {where } \mathrm{f}}$ is discontinuous, and Q denote the set containing all $x \in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to $\qquad$ [JEE Main 2021]
1) 5
2) 4
3) 2
4) 1

Solution

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

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

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

Example 5: Let $f:[0, \infty) \rightarrow[0,3]{\text {be a function }}$ defined by $f(\mathrm{x})= \begin{cases}\max \{\sin t: 0 \leq \mathrm{t} \leq \mathrm{x}\}, & 0 \leq \mathrm{x} \leq \pi \\ 2+\cos \mathrm{x}, & \mathrm{x}>\pi\end{cases}$ Then 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.

Frequently Asked Questions (FAQs)

1. What is derivative?

The instantaneous rate of change of a function concerning the independent variable is called the derivative.

2. What is differentiation?

The process of finding the derivative of a function is called differentiation.

3. What is the condition for differentiation?

A function $f(x)$ is not differentiable at $x=a$ if the function is discontinuous at $\mathrm{x}=\mathrm{a}$.

4. When a function is said to be differentiable?

A function $f(x)$ is said to be differentiable if $R f^{\prime}\left(x_{\circ}\right)$ and $L f^{\prime}\left(x_{\circ}\right)$ both exist.

5. How is derivative denoted?

Derivative of f at c is denoted by $\mathrm{f}^{\prime}(\mathrm{c})$ or $\left.\frac{d}{d x}(f(x))\right|_c$.

6. What does it mean for a function to be differentiable at a point?

A function is differentiable at a point if it has a well-defined derivative at that point. This means the function has a unique tangent line at that point, and the rate of change of the function is defined and continuous in the immediate neighborhood of that point.

7. How can you visually determine if a function is differentiable at a point using its graph?

To visually determine if a function is differentiable at a point, look for a smooth, continuous curve at that point. The graph should not have any sharp corners, cusps, or vertical tangent lines at the point in question. You should be able to draw a unique tangent line at that point.

8. What's the relationship between continuity and differentiability?

Differentiability implies continuity, but continuity does not imply differentiability. In other words, if a function is differentiable at a point, it must be continuous at that point. However, a function can be continuous at a point without being differentiable there.

9. Can a function be continuous but not differentiable at a point? Give an example.

Yes, a function can be continuous but not differentiable at a point. A classic example is the absolute value function f(x) = |x| at x = 0. The function is continuous at x = 0, but it has a sharp corner there, making it non-differentiable at that point.

10. What does a "corner" in a graph indicate about differentiability?

A corner in a graph indicates a point where the function is not differentiable. At a corner, the left-hand and right-hand derivatives exist but are not equal, meaning there's no unique tangent line at that point.

11. How can you use the concept of secant lines to visually approximate the derivative of a function at a point?

To visually approximate the derivative at a point, draw secant lines through that point and another point on the curve, gradually moving the second point closer to the first. The limiting position of these secant lines as they approach the tangent line gives an approximation of the derivative.

12. How does the graph of a function's second derivative relate to the differentiability of the original function?

The graph of a function's second derivative provides information about the rate of change of the first derivative. Discontinuities in the second derivative's graph indicate points where the original function's first derivative is not differentiable, but the original function itself may still be differentiable at these points.

13. What does the graph of a function look like near a point where it's differentiable but not twice differentiable?

Near a point where a function is differentiable but not twice differentiable, the graph will have a well-defined tangent line, but the curvature of the graph may change abruptly. The function might have a "corner" in its first derivative graph at this point.

14. How does a vertical tangent line affect the differentiability of a function?

A vertical tangent line indicates that the function is not differentiable at that point. This is because the slope of a vertical line is undefined (infinite), and therefore, the derivative at that point does not exist.

15. What is the significance of a "cusp" in a graph when examining differentiability?

A cusp in a graph indicates a point where the function is not differentiable. At a cusp, the function approaches the point from both sides with infinite slope, and there's no unique tangent line at that point.

16. How can you use the concept of left-hand and right-hand derivatives to determine differentiability?

A function is differentiable at a point if both the left-hand and right-hand derivatives exist and are equal at that point. If they exist but are not equal, or if either one doesn't exist, the function is not differentiable at that point.

17. What does a "jump discontinuity" in a graph tell you about the function's differentiability?

A jump discontinuity in a graph indicates that the function is not differentiable at that point. In fact, it's not even continuous at that point, which is a prerequisite for differentiability.

18. How does the smoothness of a curve relate to its differentiability?

The smoothness of a curve is directly related to its differentiability. A smooth curve without any abrupt changes in direction is generally differentiable at all points. The smoother the curve, the more times it can be differentiated.

19. Can a function be differentiable at a point where its graph has a horizontal tangent line?

Yes, a function can be differentiable at a point where its graph has a horizontal tangent line. In this case, the derivative at that point is simply zero. A horizontal tangent line does not affect differentiability.

20. What does it mean when we say a function is "everywhere differentiable"?

When we say a function is "everywhere differentiable," we mean that the function is differentiable at every point in its domain. The graph of such a function would be smooth without any corners, cusps, or discontinuities.

21. How does the graph of a function behave near a point where it's not differentiable?

Near a point where a function is not differentiable, the graph may exhibit one of several behaviors: it might have a sharp corner, a cusp, a vertical tangent line, or a discontinuity. The key is that there's no unique tangent line at that point.

22. Can a function be differentiable at a point where its graph crosses itself?

No, a function cannot be differentiable at a point where its graph crosses itself. At such a point, there would be multiple tangent lines, violating the definition of differentiability which requires a unique tangent line.

23. How does the differentiability of a piecewise function relate to its graph?

For a piecewise function to be differentiable at the point where the pieces meet, the graph must be continuous at that point and the left-hand and right-hand derivatives must be equal. Visually, this means the graph should have no corner or discontinuity at the joining point.

24. What can you conclude about a function's differentiability if its graph has no sharp turns or breaks?

If a function's graph has no sharp turns or breaks, it's likely differentiable at all points. However, you should still check for less obvious features like cusps or vertical tangent lines that might affect differentiability.

25. How does the concept of one-sided limits relate to examining differentiability using a graph?

One-sided limits are crucial in examining differentiability. For a function to be differentiable at a point, both one-sided limits of the difference quotient must exist and be equal. Graphically, this means the curve approaches the point smoothly from both sides.

26. Can a function with a vertical asymptote be differentiable at the point of the asymptote?

No, a function with a vertical asymptote is not differentiable at the point of the asymptote. In fact, the function is not even defined at that point, which is a prerequisite for differentiability.

27. How does the graph of a function's derivative relate to the differentiability of the original function?

The graph of a function's derivative provides information about the original function's rate of change. Discontinuities or undefined points in the derivative's graph indicate points where the original function is not differentiable.

28. What does it mean graphically for a function to have a removable discontinuity, and how does this affect differentiability?

A removable discontinuity appears as a hole in the graph of a function. While the function is not continuous (and therefore not differentiable) at this point, it could be made differentiable by redefining the function at that single point to fill the hole.

29. How can you use the intermediate value theorem to reason about a function's differentiability?

The intermediate value theorem applies to continuous functions. If a function is differentiable, it must be continuous. Therefore, if you can show that a function violates the intermediate value theorem, you can conclude it's not differentiable everywhere.

30. What does the graph of an absolutely continuous function tell you about its differentiability?

An absolutely continuous function is differentiable almost everywhere. Its graph will be smooth for the most part, with at most a countable number of points where it might not be differentiable.

31. How does the concept of local extrema relate to differentiability when examining a graph?

At a local extremum (maximum or minimum), if the function is differentiable, its derivative must be zero. Graphically, this appears as a point where the tangent line is horizontal. However, not all points with horizontal tangent lines are extrema.

32. Can a function be differentiable if its graph has a point of inflection?

Yes, a function can be differentiable at a point of inflection. In fact, points of inflection are often characterized by the second derivative changing sign, which implies the existence of the first derivative.

33. How does the graph of a function with a jump discontinuity differ from one with a removable discontinuity in terms of differentiability?

A function with a jump discontinuity is not differentiable at the point of discontinuity and cannot be made differentiable by redefining the function at that point. A function with a removable discontinuity could potentially be made differentiable by appropriately defining the function at the point of discontinuity.

34. What can you infer about a function's differentiability if its graph is a straight line?

If a function's graph is a straight line, the function is differentiable at every point in its domain. The derivative will be constant and equal to the slope of the line.

35. How does the graph of a function behave near a point where it's differentiable but its derivative is discontinuous?

Near a point where a function is differentiable but its derivative is discontinuous, the graph of the function will be continuous and have a tangent line, but the rate of change of the slope will abruptly change at that point.

36. What does it mean graphically for a function to have a vertical tangent line, and how does this affect differentiability?

A vertical tangent line appears as a vertical line touching the graph at a single point. At such a point, the function is not differentiable because the slope of the tangent line is undefined (infinite).

37. What can you conclude about a function's differentiability if its graph is composed entirely of straight line segments?

A function whose graph is composed entirely of straight line segments is differentiable at all points except where the line segments meet (assuming they meet at an angle). At these junction points, the function will have corners and thus not be differentiable.

38. How does the graph of a function with a removable discontinuity differ from one that's differentiable everywhere?

A function with a removable discontinuity will have a "hole" in its graph at the point of discontinuity, while a function that's differentiable everywhere will have a completely smooth, unbroken curve without any holes, corners, or cusps.

39. What does the graph of a function look like near a point where it's continuous but not differentiable?

Near a point where a function is continuous but not differentiable, the graph might exhibit a sharp corner, a cusp, or a vertical tangent line. The key is that there's no unique tangent line at that point, despite the function being continuous.

40. How can you use the concept of one-sided derivatives to examine differentiability graphically?

To examine one-sided derivatives graphically, look at how the function approaches a point from the left and right sides separately. If the slopes of the tangent lines from both sides are equal, the function is likely differentiable at that point.

41. What does the graph of a function look like if it's differentiable at all points except for a countable number of points?

A function that's differentiable at all points except for a countable number of points will have a graph that's mostly smooth, with isolated points where there might be corners, cusps, or other non-differentiable features.

42. How does the concept of Lipschitz continuity relate to differentiability when examining a graph?

Lipschitz continuity implies that a function doesn't change too rapidly. Graphically, a Lipschitz continuous function won't have any vertical tangent lines. While Lipschitz continuity doesn't guarantee differentiability, it does rule out certain types of non-differentiable behavior.

43. What can you infer about a function's differentiability if its graph is symmetric about the y-axis?

Symmetry about the y-axis doesn't directly imply anything about differentiability. The function could be differentiable everywhere, or it could have points of non-differentiability that are symmetrically placed about the y-axis.

44. How does the graph of a function behave near a point where it's infinitely differentiable?

Near a point where a function is infinitely differentiable, the graph will be extremely smooth. You should be able to zoom in indefinitely and the curve will always appear smooth, without any abrupt changes in direction.

45. What does the graph of a nowhere differentiable continuous function look like?

A nowhere differentiable continuous function, like the Weierstrass function, has a graph that's continuous but extremely jagged. No matter how much you zoom in, you'll never see a smooth curve or a clear tangent line at any point.

46. How can you use the concept of directional derivatives to examine differentiability of multivariate functions graphically?

For multivariate functions, examine the surface in different directions from a point. If the directional derivatives exist and are consistent (forming a plane) in all directions, the function is likely differentiable at that point.

47. What can you conclude about a function's differentiability if its graph has a cusp pointing upward?

If a function's graph has a cusp pointing upward, the function is not differentiable at the cusp point. At a cusp, the left-hand and right-hand derivatives exist but are not equal (both are typically infinite for an upward-pointing cusp).

48. What does it mean graphically for a function to have a "kink", and how does this affect differentiability?

A "kink" in a function's graph appears as a sharp change in direction, similar to a corner but possibly less pronounced. At a kink, the function is typically continuous but not differentiable, as the left-hand and right-hand derivatives exist but are not equal.

49. How can you use the concept of implicit differentiation to examine the differentiability of curves defined by equations?

Implicit differentiation allows us to find the derivative of y with respect to x for curves defined by equations like f(x,y) = 0. Graphically, points where this derivative is undefined (e.g., division by zero) indicate potential issues with differentiability.

50. What can you infer about a function's differentiability if its graph is a perfect circle?

A perfect circle is differentiable at every point except possibly where it's defined piecewise (e.g., at θ = 0 in polar coordinates). The derivative exists and is continuous everywhere on the circle, representing the changing slope of the tangent line.

51. How does the concept of smoothness classes (C^k functions) relate to what we can observe in a function's graph?

The smoothness class of a function relates to how many times it can be continuously differentiated. Graphically, higher smoothness classes (larger k) correspond to curves that appear increasingly smooth, even under magnification.

52. How can you use the mean value theorem to reason about a function's differentiability from its graph?

The mean value theorem states that for a continuous function on a closed interval, differentiable on the open interval, there's at least one point where the tangent line is parallel to the secant line of the endpoints. Graphically, this helps us understand the relationship between the function's values and its rate of change.

53. What can you conclude about a function's differentiability if its graph is self-similar (fractal-like) at all scales?

If a function's graph is truly self-similar at all scales (fractal-like), it's likely not differentiable at any point. True mathematical fractals, like the Koch snowflake, are continuous everywhere but differentiable nowhere.

54. How does the graph of a function with jump discontinuities in its derivative differ from one that's everywhere differentiable?

A function with jump discontinuities in its derivative will have points where the graph abruptly changes direction, though without forming a sharp corner. An everywhere differentiable function, in contrast, will have a smooth graph without any such abrupt changes.

55. What does the graph of a function look like near a point where it's differentiable, but its derivative is not continuous?

Near a point where a function is differentiable but its derivative is not continuous, the graph will have a well-defined tangent line, but the rate at which the tangent line changes direction will abruptly shift at that point. The curve itself will be continuous and smooth, but its curvature will change suddenly.