Introduction
The derivative, measures the rate at which the dependent variable changes with respect to the independent variable. It is one of the most important ideas in Calculus. The differentiation of functions are widely used in science, economics, medicine and computer science.
Derivative at a Point
Let f be a function and a be any point in its domain. Let h>0 be a small number.
f(x) is said to be differentiable if 
exists and is denoted by f|(a)
Interpretation of Derivative at a Point
Physical Significance: Let Q (t) be a quantity that changes with time 't'.
Let Dt be the increment given to 't' and DQ be the corresponding increment in Q, then 
is called the average rate of change of Q with respect to 't'. (Also known as Newton - quotient of f at t).
Derivative of a Function (in general)
The derivative of the function f with respect to a variable x is the function f ' whose value at x is
provided the limit exists.
Differentiability
We have already defined the derivative of a function f(x) at a particular point 'a' and derivative of f(x) in general for the variable x as f'(a) and f'(x) respectively. The restriction in both the cases is that 'the limit must exist'. If
does not exist, then we say that the function is not differentiable.
Derivative of Some Important Functions
The Derivative of Some Important Functions are: 1. Derivative of a Constant, 2. Derivative of xn where n is any integer, 3. Derivative of a Constant of a Function, 4. Derivative of Exponential Function.
Product Rule for Differentiation
'Derivative of the product of two functions =
first function x derivative of second function + second function x derivative of first function'
Quotient Rule for Differentiation
The Quotient Rule for Differentiation is:
Derivative of a Function of a Function
The differentiation of function of a function is known as 'chain rule'. The chain rule is probably the most widely used differentiation rule in mathematics.
If y is a differentiable function of u and u is a differentiable function of x, then the derivative of y with respect to x is equal to the derivative of y with respect to u times the derivative of u with respect to x.
Derivative of Inverse Trignometric Functions
The Derivative of Inverse Trignometric Functions includes: 1. sin-1x, 2. cos-1x, 3. tan-1x, 4. cot-1x, 5. sec-1x, 6. cosec-1x.
Derivative of Implicit Functions
Till now, the functions that we have discussed, are explicitly functions of x. We have defined y in terms of x. Suppose we have an equation f(x,y) = 0, which cannot be put in the form of y=f(x) to differentiate in the usual way, we can still differentiate the equation f(x,y) = 0. This function in which y cannot be expressed in terms of x, is called an implicit function.
Logarithmic Differentiation
When we want to differentiate a function of the form f(x) g(x), we use logarithmic differentiation.
Let y = f(x) g(x)
Taking log on both sides, we have logy = g(x) logf(x).
Differentiating with respect to x, we get,
Differentiation of Parametric Functions
If x=f(t) and y=g(t), where x and y are dependant on the independent variable t, then
t is called the parameter.
Differentiation by Substitution
Differentiation of certain functions seem to be very difficult, but by suitably substituting the independent variable with some trigonometric function or other functions, they can be differentiated easily.
If f(x) involves inverse trigonometric functions of algebraic functions, the substitutions simplify the function f(x) to be differentiated.
Derivatives of Second Order
The order of a differential equation is equal to the highest derivative in the equation. The single-quote indicates differentiation. So x' is a first derivative, while x'' is a second derivative.
x' = 1/x is first-order.
x'' = -x is second-order
x'' + 2 x' + x = 0 is second-order.
Summary
1. f'(a) exists at x = a iff Lf' (a) = Rf'(a). Where Rf '(a) is Right hand derivative and Lf '(a) is Left hand derivative.
2. Derivability implies continuity.
3. Derivative of a constant function is zero.
Conclusion
We have studied various techniques of differentiation. Also, we have studied the method of obtaining higher order derivatives of functions which is useful in maxima and minima problems.
