#### notes

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.

#### theorem

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.