Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Differentiation

Instantaneous rate of Change and Differentiability

When a quantity undergoes change, the change in the quantity is always associated, in almost every field of human endeavor, with the rate at which the quantity change. The rate of change in the quantity may be observed with respect to another quantity such as time, distance, or position in space, etc.

The instantaneous rate of change, i.e. the rate at which the quantity changes at any particular time. If a person travels at a uniform speed. i.e. if there is no change in the rate at which he is changing his position, then the instantaneous rate of change of position at any point would remain a constant.

Differentiability

If  f is a function whose graph is a straight line then  \dfrac{f(\text {t}) - f (\operatorname{t_0})}{\operatorname{t - t_0}} is nearly a constant when  \operatorname{t-t_0} is small. i.e. there is a constant on  m such that  \dfrac{f(\text{t}) - f (\operatorname{t_0})}{\operatorname{t - t_0}} = m + r , where  r  can be made small by taking  t close enough to \operatorname{t_0} , this means,  \lim_{t \to t_o} \dfrac{f(\text{t}) - f (\operatorname{t_0})}{\operatorname{t - t_0}} = m (Differentiability of  t  at  \operatorname{t_0} .)

Definition of Derivatives

A function f : (a, b) \longrightarrow R is said to be derivable or differentiable at  x \in (a, b) if   \lim_{\delta x \to 0} \dfrac{\delta y}{\delta x} = \lim_{\delta x \to 0} \dfrac{f(x + \delta x) - f(x)}{\delta x} exists.

This limit is called the derivative of  f with respect to (w. r. t)  x .The derivative of  f is denoted by  f' or  \dfrac{dy}{dx} or sometimes  D_y or  D_f .

We can also write,

Let  C \in (a, b) and  f be derivable at  C , then

 \dfrac{dy}{dx} \bigg]_{x = c} = f'(C) = \lim_{h \to 0}\quad \dfrac{f(c+h) - f(1)}{h}

Left-hand derivative:

When  h \to o^- (i.e.  h \to 0 through negative values only) the limit is called the left- hand derivative of  f at  C and is denoted by  f'(C^-) .

Thus  f'(C^-) = \lim_{h \to o} \quad \dfrac{f(C-h) - f(c)}{-h}, h> 0 .

Right-hand derivative

When  h \to 0^+ (i.e. when  h \to 0 through positive values only) the limit is called the right hand derivative of  f at  C and is denoted by  f'(C^+) .

Thus  f'(C^+) = \lim_{h \to o} \quad \dfrac{f(C + h) - f(C)}{h}, h > 0

Note:

It is easily seen that derivative of   f(x) exists at   x=C iff  f'(C^+) and  d'(C^-) both exists and are equal and irrespective of sign of  h .

If  f has a derivative at every point of the interval  (a, b) , then it is said to be differentiable on  (a,b) . The process of finding the derivative of a function is known as differentiation.

 

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