Derivative of a Function: Differential and Differentiability of a Function

The derivative of a function at a number ‘a’ is denoted by f’(a) and is defined by

f′(a)=lim_h→0 [(f(a+h)-f(a)) / h]

if this limit exists.

If we assume that x=a+h then x→a as h→0 and we can write definition of derivative of a function in this way

f’(a)=lim_x→a [(f(x)-f(a)) / (x-a)]

Where f(x)=f(a+h) and (x-a)=h.

Derivative of a function
Derivative of a Function

Interpretation of the Derivative as the Slope of a Tangent

The derivative of a function y=f(x) denoted by y’=f’(x) at a point x=a is the Slope of a Tangent line to y=f(X) at the given point x=a.

In other words, the tangent line to y=f(x) at a point x=a or (a, f(a)) is the line through y=(a, f(a)) whose slope is equal to f’(a), the derivative of function at x=a.

Differential of a Function

The differential of a function Y=f(x) is denoted by dy and is defined as;

dy = f’(x) dx

Where f'(x) represents the derivative of the function f(x) with respect to x, and dx represents an infinitesimally small change in the independent variable x. The differential dy represents the corresponding small change in the dependent variable Y due to the change in x.

For Example;

Let’s consider a function f(X)=X². 

The derivative of f(X) with respect to X is f’(X)=2X.

By definition of differential of a function y=f(X),

dy = f’(X) dX

if we take dx to be a small change in x, we can find the corresponding change dy in y=f(X).

Let’s say x=3 and dx=0.1, using value x=3 in f’(X)=2X we get 

f’(3) = 2×3 = 6

Now by definition

dy = f’(3) dX = 6 × 0.1 = 0.6 

Therefore, for f(x) = X² at X= 3 with dx = 0.1, a small change in X results in a corresponding change dy = 0.6 in y=f(X).

Differentiable Function

A function Y=f(x) is said to be differentiable at a point x=a if f’(x), the derivative of f(X) exists at x=a. It is differentiable at an open interval (a, b) if it is differentiable at every point in the interval.

For Example;

Check where is the function f(X)=|X| differentiable?


If X>0 then |X|=X, and we take h small as much that X+h>0 then |X+h|=(X+h).

By definition of derivative of a  function for X>0, we have

f′(X) = lim_h→0 [(f(X+h)-f(X)) / h] = lim_h→0 [((X+h)-X) / h] = lim_h→0 [ h/ h] = lim_h→0 (1) = 1.

So f(X) = |X| is differentiable for each X>0.

If X<0 then |X|=(-X), and we take h small as much that X+h<0 then |X+h| = -(X+h).

By definition of derivative of a  function for X<0, we have

f'(X) = lim_h→0 [(f(X+h)-f(X)) / h] = lim_h→0 [(-(X+h)-(-X)) / h] = lim_h→0 [ -h/ h] = lim_h→0 (-1) = -1.

So f(X) = |X| is differentiable for each X<0.

If X=0, then |X|=|0|=0 . In this case, since function f(X)=|X| is changing its concavity (concave upward to concave downward or Increasing  to Decreasing) at X=0, so X=0 is the inflection point of f(X)=|X|. AS X=0 being inflection point of function is also a point at which function f(X)=|X| is divided into two parts ( f(X)=|X|=X when X>0 and f(X)=|X|=(-X) when X<0 ).

To check differentiability of a function at a point on which it is divided into pieces, we have need to check f’(X^-) and f’(X^+) values of function to the left hand side and right hand side of this point if  values are equal then function is said to be differentiable i.e 

Lim_h→0^- [((X+h)-f(X)) / h] = Lim_h→0^+ [((X+h)-f(X)) / h]

For X=0, computing left hand and right hand limits respectively,

Lim_h→0^- [(f(X+h)-f(X)) / h] = Lim_h→0^- [ (|0+h|- 0) / h] = Lim_h→0^- [ -h/ h] = Lim_h→0^- (-1) = -1

Lim_h→0^+ [(f(X+h)-f(X)) / h] = Lim_h→0^+ [ (|0+h|- 0) / h] = Lim_h→0^+ [ +h/ h] = Lim_h→0^+ (+1) = 1

Since left hand and right hand limits are not equal, so f(X) = |X| is not  differentiable at X=0.

Hence f(X)=|X| is differentiable at each point X accept X=0.

How Can a Function Fail to be Differentiable ?

There exist three scenarios where a function may fail to be differentiable:

1) Consider the function y = |x|. At x = 0, the function isn’t differentiable because its graph abruptly changes direction, creating a ‘corner’ or ‘kink.’ Whenever a function’s graph exhibits such characteristics, it lacks a tangent at that point, rendering it non-differentiable. This occurs when attempting to compute f'(a) and finding differing left and right limits.

2) According to the theorem ‘differentiability implies continuity, but the converse is not true,’ a function may lack a derivative if it’s not continuous at a point ‘a.’ Thus, any discontinuity, like a jump discontinuity, leads to the function being non-differentiable.

3) Another possibility arises when the curve has a vertical tangent line at x = a. Here, f remains continuous at ‘a,’ yet Lim_X→a|f’(x)|= ∞, indicating that tangent lines grow infinitely steep as x approaches ‘a’.

Noman Yousaf

Meet Noman Yousaf, a Math graduate from University of Education Lahore Jauharabad Campus. He excels at simplifying complex math topics, teaching with clarity and making math understandable for all.

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