Suppose f is a real function and c is a point in its domain. The derivative of f at c is defined by
`lim_(h->0) (f(c+h) - f(c))/h`
provided this limit exists. 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`
wherever the limit exists is defined to be the derivative of f. The derivative of f is denoted by f'(x) or `d/(dx)`(f(x)) or if y = f(x) by `(dy)/(dx)` or y' .
The process of finding
derivative of a function is called differentiation. We also use the phrase differentiate f(x) with respect to x to mean find f′(x).
The following rules were established as a part of algebra of derivatives:
(1) (u ± v)′ = u′ ± 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`|
Whenever we defined derivative, we had put a caution provided the limit exists.
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
`lim_(x->c) (f(x) - f(c))/(x-c) = f'(c)`
But for x ≠ c, we have
f(x) – f(c) = `(f(x) - f(c))/(x-c). (x-c)`
Therefore `lim_(x->c) [f(x) -f(c)] = lim_(x->c) [(f(x) - f(c))/(x-c) . (x-c)]`
or `lim_(x->c) [f(x)] - lim_(x->c)[f(c)] = lim_(x->c)[(f(x)-f(c))/(x-c)] . lim_(x->c) [(x - c)] `
= f′(c) . 0 = 0
or `lim _(x->c)` f(x) = f(c)
Hence f is continuous at x = c.
Shaalaa.com | Continuity and Differentiability part 15 (Algebra of Derivatives)
Discuss the continuity and differentiability of the