Concept of Differentiability

Differentiability is a core concept in calculus that determines whether a function has a well-defined derivative (or a continuous rate of change) at a particular point. The process of finding this derivative is called differentiation.

The derivative of a real function f at a point c in its domain is defined as:

\[f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}\]

If u and v are differentiable functions, then:

• \[(u \pm v)' = u' \pm v'\]

• \[(uv)' = u'v + uv'\]

• \[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}, v \neq 0\]

• \[\frac{d}{dx}(x^n) = nx^{n-1}\]

• \[\frac{d}{dx}(\sin x) = \cos x\]

• \[\frac{d}{dx}(\cos x) = -\sin x\]

• \[\frac{d}{dx}(\tan x) = \sec^2 x\]

A function is differentiable at \[x = c\] if both one-sided derivatives exist, are finite, and are equal.

Left-hand derivative:

\[ \boxed{\lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h}} \]

Right-hand derivative:

\[ \boxed{\lim_{h \to 0^+} \frac{f(c+h) - f(c)}{h}} \]

Thus, \[ \boxed{\text{LHD} = \text{RHD}} \]

is the practical condition for differentiability.

A function is differentiable on an open interval (a,b) if it is differentiable at every point of (a,b).

For a closed interval [a,b][a,b]:

• at a, the right-hand derivative is considered
• at b, the left-hand derivative is considered

If a function \[f\] is differentiable at a point \[c\], then it is also continuous at that point.

Proof: Since \[f\] is differentiable at \[c\], we have

But for \[x \neq c\], we have

Therefore \[\lim_{x \to c} [f(x) - f(c)] = \lim_{x \to c} \left[ \frac{f(x) - f(c)}{x - c} \cdot (x - c) \right]\]

or \[\lim_{x \to c} [f(x)] - \lim_{x \to c} [f(c)] = \lim_{x \to c} \left[ \frac{f(x) - f(c)}{x - c} \right] \cdot \lim_{x \to c} [(x - c)]\]

 \[= f'(c) \cdot 0 = 0\]

\[ \boxed{\lim_{x \to c} f(x) = f(c)} \]

Hence \[f\] is continuous at \[x = c\].

Converse is Not True

A continuous function need not be differentiable.

Consider \[ f(x) = |x|. \]

At \[x = 0\], \[ \text{LHD} = \lim_{h \to 0^-} \frac{|h| - 0}{h} = -1 \]

while \[ \text{RHD} = \lim_{h \to 0^+} \frac{|h| - 0}{h} = 1. \]

Since \[ -1 \neq 1, \]

\[ \boxed{|x|\ \text{is not differentiable at}\ x = 0} \]

although it is continuous there.

• Derivative exists only when the defining limit exists.

• Differentiability at a point means the function has a valid derivative there.

• Every differentiable function is continuous at that point.

• Every continuous function is not necessarily differentiable.