Mathematics Differentiablity, Derivative of composite function and Chain Rule For CBSE-NCERT
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### Topic covered

star Differentiability
star Theorem : If A function f is differentiable at a point c, then it is also continuous at that point.
star Derivatives of composite functions
star Theorem : (Chain Rule)

### Differentiability

● If f is a real function and c is a point in its domain. The derivative of f at c is defined by color{red}{lim_(h-> 0) (f(c+h)-f(c))/h}

●So a function f is differentiable at a point c in its domain if both lim_(h-> 0^-) (f(c+h)-f(c))/h and lim_(h-> 0^+) (f(c+h)-f(c))/h are finite and equal.

color{blue}{lim_(h-> 0^-) (f(c+h)-f(c))/h = lim_(h-> 0^+) (f(c+h)-f(c))/h}

●Derivative of f at c is denoted by f '(c) or d/(dx) (f(x))|_c The function defined by

f'(x) =lim_(h->0) (f(x+h)-f(x))/h

- The derivative of f is denoted by f ' (x) or d/(dx) (f(x)) or y' f (x) with respect to x to mean find f '(x).

The following rules of algebra of derivatives :

(1) (u pm v)' = u' pm v'

(2) (uv)'= u'v + uv' (Leibnitz or product rule)

(3) (u/v)' =((u'v -uv')/v^2) wherever v != 0 (Quotient rule).

The following table gives a list of derivatives of certain standard functions:

 f(x) x^n sin x cos x tan x f'(x) nx^(n-1) cos x -sin x sec^2 x

Differentiation formulas list :