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 :

Theorem : If A function f is differentiable at a point c, then it is also continuous at that point.