Mathematics Revision Notes of Differentiablity for NDA

Differentiability :

Let `f` be a real valued function defined on an open interval `(a, b)` and `c in (a, b)`. Then, `f(x)` is said to be differentiable or
derivable at `x = c`, if `lim_(x-> c) (f(x) -f(c))/(x-c)` exists finitely.

This limit is called the derivative or differential coefficient of the function `f(x)` at `x = c` and is denoted by `f'(c)` or `D f(c)` or `| d/(dx) f(x) |_(x=c)` If we vary the point cover all numbers for which the limit, `lim_(x-> c) (f(x)-f(c))/(x-c)` exists, then we obtain `f'` or `f' (x)` and it is called derivative of `f`.

Let `f(x)` be a function and c be a point in the domain of `f`.

Then, the right hand derivative (RHD) denoted by `Rf'(c)` is defined as

`RHD = Rf'(c) = lim_(x-> c^(+)) (f(x) -f(c))/(x-c)`

`RHD =lim_(h->0) (f(c+h) -f(c))/h`

and the left hand derivative (LHD) denoted by `Lf'(c)` is defined as

`LHD =Lf'(c) =lim_(x->c^(-)) (f(x) -f(c))/(x-c)`

`LHD = lim_(h->0) =(f(c-h)-f(c))/(-h)`

A function `f(x) ` is said to be differentiable at a point `c` in its domain, if

Left hand derivative = Right hand derivative

or `LHD =RHD`

i.e., `LF'(c) = Rf' (c)`

`=> lim_(x-> c^(-)) (f(x) -f(c))/(x-c) = lim_(x->c^(+)) (f(x) -f(c))/(x-c)`

If `Lf'(c) ne Rf(c)`, then function `f(x)` is not differentiable at `x =c`.

NOTE
Every differentiable function is continuous..
But the converse of this statement is not true, i.e. every continuous function may not be differentiable.
Geometrically, f(x)is differentiable at a point P, if the curve does not have Pas a corner point, i.e. the function is not
differentiable at those points on which function has jumps (or holes) and sharp edges.

How can a function fail to be differntiable :

(a) The function f(x) is said to non-differentiable at `x = a` if Both left and tight and hand derivative exists but are not equal

The function `y = |x|` is not differentiable at `0` as its graph change direction abrupty when `x = 0`. In general, if the graph of a function
has a 'corner' or 'kink' in it, then the graph of `f` has no tangent at this point and this not differentiable there. (To compute `f' (a)`, we find that the left and tight limits are different.)

(b) Function is discontinuous at `x =a`

If `f` is not continuous at a then `f` is not differentiable at `a`. So at any discontinuity (for instance, a jump of discontinuity) f fai ls to be
differentiable.

(c) Either or both left and tight hand derivative are not finite.

A third possibility is that the curve has a vertical tangent line when `x = a`, that is `f` is continuous at `a` and `lim_(x-> a) f' ( x) = oo`.

This means that the tangent lines becomes steeper and sleeper as `x =a`.

Differentiability in an Interval :

A function `y = f(x)` defined on an open interval `(a, b)` is said to be differentiable in an open interval (a, b), if it is
differentiable at each point of `(a, b)`.

A function `y = f(x)` defined on a closed interval `[a, b]` is said to be differentiable in closed interval `[a, b]`, if it is differentiable at each point of an open interval `(a, b)` and `lim_(x-> a^+) (f(x) -f(a))/(x-a) ` and `(f(x) -f(b))/(x-b)` both exist A function f is said to be differentiable, if it is differentiable at every point of its domain.

Some Standard Results on Differentiability :

(i) Every polynomial function, every exponential function `a^x (a > 0)` and every constant function are
differentiable at each `x in R`.

(ii) The logarithmic functions, trigonometrical functions and inverse trigonometrical functions arc always
differentiable in their domains.

(iii) The composite of differentiable function is a differentiable function.

(iv) Absolute functions are always continuous throughout but not differentiable at their critical
points.

Points to Remember :

lf `f(x)` and `g (x)` both are derivable at `x = a`, then

(i) `f(x) ± g(x)` will be differentiable at `x = a`.

(ii) `f(x) * g (x)` will be differentiable at `x = a`.

(iii) `(f(x))/(g(x))` will be differentiable at `x = a` if `g(a) ne 0`.

Note that :

(1) lf `f(x)` and `g (x)` are both derivable at `x = a, f(x) ± g (x); g(x) f (x)` and `(f(x))/(g(x))` will also be derivable at `x = a`. (only if `g (a) ne 0`)

(2) lf `f(x)` is derivable at `x = a` and `g (x)` is not derivable at `x = a` then the `f(x) + g (x)` or `f(x)- g (x)` will not
be derivable at `x = a`.

(3) Derivative of a continuous function need not be a continuous function

Comparision of Limit Continuity and Differentiabilility in Different cases :

(1) If Both Limit `f(x)` and `g(x)` exist ,

sum , difference , product, division (when `D^r ne 0` ) will exist.

(2) If One exist and One doesn't.

sum and difference will not exist

Nothing can be said about existance of product and division.

(3) If Both Don't exist

Nothing can be said about sum, difference , product and division.

(4) If product, sum, difference, division exist then nothing can be said about existance of individual limits.

Some Standard Results On Differentiability :

` tt ((, " Function "f(x) , "Intervals in which "f(x) "is differentiable"),
(1. , "Polynomial ", (-oo,oo)),
(2., "Exponential "(a^x, a > 0) , (-oo,oo)),
(3., "Constant" , (-oo, oo)),
(4., "Logarithmic " , " Each point in its domain "),
(5., " Trigonometric " , "Each point in its domain"),
(6., "Inverse trigonometric" , "Each point in its domain"))`