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The derivative of a real function f at point c in its domain is defined by:

$\underset{h\xe2\u2020\u2019\text{0}}{\text{lim}}\frac{\text{f(c+h)-f(c)}}{\text{h}}$

Derivative of f(x) at c = f '(c) or d/dx[f(x)]_{c}

f ' (x) = $\underset{h\xe2\u2020\u2019\text{0}}{\text{lim}}\frac{\text{f(x+h)-f(x)}}{\text{h}}$ provided the limit exists.

Let y = f(x)

The derivative of y is denoted by y ' or dy/dx.

Differentiation is the process of finding the derivative of a function

If the limit does not exist, it means that function f is not differentiable at c.

A function *f* is said to be differentiable at point *c* in its domain, if:

$\underset{h\xe2\u2020\u2019{\text{0}}^{\text{-}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$ = $\underset{h\xe2\u2020\u2019{\text{0}}^{\text{+}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$

A function *f* is said to be differentiable in interval [a,b], if it is differentiable at every point of [a,b].

*f ' *(a) is called the right hand derivative and *f '* (b) is called the left hand derivative of *f*.

A function is said to be differentiable in (a,b), if it is differentiable at every point of (a,b).

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

**Proof:**

Given function *f* is differentiable at point *c*.

By definition, f ' (c) = $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\frac{\text{f(x) - f(c)}}{\text{x - c}}$

For *x â‰ c*, we have

*f(x) - f(c) = (f(x) - f(c))/(x - c) x (x - c)*

$\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{[f(x) - f(c)]}$= $\underset{x\xe2\u2020\u2019c}{\text{lim}}\text{[}\frac{\text{f(x) - f(c)}}{\text{x - c}}\text{}\xc3\u2014\text{(x - c)]}$

â‡’ $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(x) -}$$\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(c)}$$$= $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{[}\frac{\text{f(x) - f(c)}}{\text{x - c}}\text{]}$$\text{\xc3\u2014}$$\text{}$ $\underset{x\xe2\u2020\u2019c}{\text{lim}}\text{(x - c)}$

= f ' (c) = 0

â‡’ $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(x)}$ = $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(c)}$ = f(c)

Hence, *f* is continuous at *x = c*.

Note:

All differentiable functions are continuous, but the converse is not always true.

The derivative of a real function f at point c in its domain is defined by:

$\underset{h\xe2\u2020\u2019\text{0}}{\text{lim}}\frac{\text{f(c+h)-f(c)}}{\text{h}}$

Derivative of f(x) at c = f '(c) or d/dx[f(x)]_{c}

f ' (x) = $\underset{h\xe2\u2020\u2019\text{0}}{\text{lim}}\frac{\text{f(x+h)-f(x)}}{\text{h}}$ provided the limit exists.

Let y = f(x)

The derivative of y is denoted by y ' or dy/dx.

Differentiation is the process of finding the derivative of a function

If the limit does not exist, it means that function f is not differentiable at c.

A function *f* is said to be differentiable at point *c* in its domain, if:

$\underset{h\xe2\u2020\u2019{\text{0}}^{\text{-}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$ = $\underset{h\xe2\u2020\u2019{\text{0}}^{\text{+}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$

A function *f* is said to be differentiable in interval [a,b], if it is differentiable at every point of [a,b].

*f ' *(a) is called the right hand derivative and *f '* (b) is called the left hand derivative of *f*.

A function is said to be differentiable in (a,b), if it is differentiable at every point of (a,b).

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

**Proof:**

Given function *f* is differentiable at point *c*.

By definition, f ' (c) = $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\frac{\text{f(x) - f(c)}}{\text{x - c}}$

For *x â‰ c*, we have

*f(x) - f(c) = (f(x) - f(c))/(x - c) x (x - c)*

$\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{[f(x) - f(c)]}$= $\underset{x\xe2\u2020\u2019c}{\text{lim}}\text{[}\frac{\text{f(x) - f(c)}}{\text{x - c}}\text{}\xc3\u2014\text{(x - c)]}$

â‡’ $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(x) -}$$\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(c)}$$$= $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{[}\frac{\text{f(x) - f(c)}}{\text{x - c}}\text{]}$$\text{\xc3\u2014}$$\text{}$ $\underset{x\xe2\u2020\u2019c}{\text{lim}}\text{(x - c)}$

= f ' (c) = 0

â‡’ $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(x)}$ = $\underset{x\xe2\u2020\u2019\text{c}}{\text{lim}}\text{f(c)}$ = f(c)

Hence, *f* is continuous at *x = c*.

Note:

All differentiable functions are continuous, but the converse is not always true.