FIRST PRINCIPLE OF DIFFERENTIATION The derivative of a given function f at a point x = a on its domain is defined as: f (a h) f (a) provided the limit exists & is denoted by f (a). Lim h 0 h f ( x) f (a) f (a) = Lim , provided the limit exists. x a x a If x and x + h belong to the domain of a function f defined by y = f(x), then f ( x h) f ( x) Lim if it exists, is called the Derivative of f at x & h 0 h is denoted by f (x) or dy . i.e., f ( x h) f ( x) dx f (x) = Lim h 0 h This method of differentiation is also called ab-initio method or first principle.
EXAMPLESFind derivative of following functions by first principle(i) f(x) = x2 (ii) f(x) = tan x(iii) f(x) = esinx Ans. (i) 2x (ii) sec2x (iii) cos x .esinx
DIFFERENTIATION OF SOME ELEMENTARYFUNCTIONS f(x) f (x) xn nxn-1 ax ax ln a ln |x| 1/x logax 1/(x ln a) sin x cos x cos x – sin x sec x sec x tan x cosec x – cosec x cot x tan x sec2 x cot x – cosec x
BASIC THEOREMS D (f ± g) = f (x) ± g (x) D (k f(x)) = k D f(x) D (f(x) . g(x)) = f(x) g (x) + g(x) f (x) D (f(g(x))) = f (g(x)) g (x) f (x) g( x ) f ( x ) f ( x ) g ( x ) D = g( x ) g2 ( x ) This rule is also called the chain rule of differentiation and can be written as dy dy dz . dx dz dx
Note that an important inference obtained from the chain rule is that dy dy dx 1 . dy dx dy dy 1 dx dx / dy another way of expressing the same concept is by considering y = f(x) and x = g(y) as inverse functions of each other. dy dx f ( x) or g ( x) dx dy 1 g ( y ) f ( x)
DERIVATIVE OF INVERSE TRIGONOMETRICFUNCTIONS y = sin–1 x – y x = sin y dx dy 1 1 cos y dy dx cos y 1 sin 2 y dy 1 ; 1 x 1 dx 1 x 2 Note here that cos y 1 sin 2 y,rather cos y 1 but2 y sin for values of y , cos y is always positive and , 2 2 hence the result. Similarly it is for other inverse trigonometric functions.
DERIVATIVE OF INVERSE TRIGONOMETRICFUNCTIONS f(x) f (x) 1 sin–1x 2 ; |x| < 1 1 x 1 cos–1x ; |x| < 1 2 1 x 1 tan–1x 2 ; x R 1 x 1 cot–1x ; x R 2 1 x 1 sec–1 x ; |x| > 1 | x | x2 1 1 ; |x| > 1 cosec-1 x | x | x2 1
METHODS OF DIFFERENTIATION Logarithmic Differentiation The process of taking logarithm of the function first and then differentiate is called Logarithmic Differentiation. It is useful if A function is the product or quotient of a number of functions OR A function is of the form [f(x)]g(x) where f & g are both derivable. Implicit differentiation If f(x, y) = 0, is an implicit function then in order to find dy/dx, we differentiate each term w.r.t. x regarding y as a functions of x & then collect terms in dy/dx.
Differentiation using substitution Following substitutions are normally used to simplify these expressions. x2 a2 x = a tan or a cot a2 x2 x = a sin or a cos 2 2 x = a sec or a cosec x a x a x = a cos a x
Parametric Differentiation If y = f( ) & x = g( ) where is a parameter, then dy dy / d dx dx / d Derivative of one function with respect to another Let y = f(x); z = g(x) then dy dy / dx f (x) dz dz / dx g(x)
DERIVATIVES OF HIGHER ORDER Let a function y = f(x) be defined on an open interval (a, b). It’s derivative, if it exists on (a, b) is a certain function f (x) [or (dy/dx) or y ] & is called the first derivative of y w.r.t. x. If it happens that the first derivative has a derivative on (a, b) then this derivative is called the second derivative of y w.r.t. x & is denoted by f (x) or (d2y/dx2) or y . Similarly, the 3rd order derivative of y w.r.t. x, if it exists, is defined by It is also denoted by f (x) or y .
f ( x ) g( x ) h( x ) If f(x) = l( x ) m( x ) n( x ) u( x ) v( x ) w( x ) where f, g, h, l, m, n, u, v, w are differentiable functions of x then f ( x ) g ( x ) h ( x ) f ( x ) g( x ) h( x ) f (x) = l( x ) m( x ) n( x ) + l ( x ) m ( x ) n ( x ) u( x ) v( x ) w( x ) u( x ) v( x ) w( x ) f ( x ) g( x ) h( x ) + l( x ) m( x ) n( x ) u ( x ) v ( x ) w ( x )
L’ HOSPITAL’S RULEIf f(x) & g(x) are functions of x such that: Lim f ( x) 0 Lim g ( x) OR Lim f ( x) Lim g ( x) x a x a x a x a Both f(x) & g(x) are continuous at x = a . Both f(x) & g(x) are differentiable at x = a . Both f (x) & g (x) are continuous at x = a, Then f ( x) f ( x) f ( x) Lim Lim Lim x a g ( x) x a g ( x ) x a g ( x ) & so on till indeterminant form vanishes.