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:

• \[(u \pm v)' = u' \pm v'\]

• \[(uv)' = u'v + uv'\]

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

• \[\frac{d}{dx}(x^n) = nx^{n-1}\]

• \[\frac{d}{dx}(\sin x) = \cos x\]

• \[\frac{d}{dx}(\cos x) = -\sin x\]

• \[\frac{d}{dx}(\tan x) = \sec^2 x\]

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

 \[= f'(c) \cdot 0 = 0\]

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

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

• 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.