## Chapter 5: Continuity and Differentiability

### Derivative

The rate of change of a quantity y with respect to another quantity x is called the derivative or differential coefficient of y with respect to x .

### Differentiation of a Function

Let f(x) is a function differentiable in an interval [a, b]. That is, at every point of the interval, the derivative of the function exists finitely and is unique. Hence, we may define a new function g: [a, b] ? R, such that, ? x ? [a, b], g(x) = f'(x).
This new function is said to be differentiation (differential coefficient) of the function f(x) with respect to x and it is denoted by df(x) / d(x) or Df(x) or f'(x).

### Differentiation from First Principle

Let f(x) is a function finitely differentiable at every point on the real number line. Then, its derivative is given by

#### Standard Differentiations

1. d / d(x) (xn) = nxn  1, x ? R, n ? R
2. d / d(x) (k) = 0, where k is constant.
3. d / d(x) (ex) = ex
4. d / d(x) (ax) = ax loge a > 0, a ? 1

#### Fundamental Rules for Differentiation

(v) if d / d(x) f(x) = ?(x), then d / d(x) f(ax + b) = a ?(ax + b)
(vi) Differentiation of a constant function is zero i.e., d / d(x) (c) = 0.

#### Geometrically Meaning of Derivative at a Point

Geometrically derivative of a function at a point x = c is the slope of the tangent to the curve y = f(x) at the point {c, f(c)}.
Slope of tangent at P = lim x ? c f(x)  f(c) / x  c = {df(x) / d(x)} x = c or f (c).

##### Different Types of Differentiable Function
###### 1. Differentiation of Composite Function (Chain Rule)

If f and g are differentiable functions in their domain, then fog is also differentiable and
(fog) (x) = f {g(x)} g (x)
More easily, if y = f(u) and u = g(x), then dy / dx = dy / du * du / dx.
If y is a function of u, u is a function of v and v is a function of x. Then, dy / dx = dy / du * du / dv * dv / dx.

###### 2. Differentiation Using Substitution

In order to find differential coefficients of complicated expression involving inverse trigonometric functions some substitutions are very helpful, which are listed below .

###### 3. Differentiation of Implicit Functions

If f(x, y) = 0, differentiate with respect to x and collect the terms containing dy / dx at one side and find dy / dx.
Shortcut for Implicit Functions For Implicit function, put d /dx {f(x, y)} =  ?f / ?x / ?f / ?y, where ?f / ?x is a partial differential of given function with respect to x and ?f / ?y means Partial differential of given function with respect to y.

###### 4. Differentiation of Parametric Functions

If x = f(t), y = g(t), where t is parameter, then dy / dx = (dy / dt) / (dx / dt) = d / dt g(t) / d / dt f(t) = g (t) / f (t)

###### 5. Differential Coefficient Using Inverse Trigonometrical Substitutions

Sometimes the given function can be deducted with the help of inverse Trigonometrical substitution and then to find the differential coefficient is very easy.

#### Logarithmic Differentiation Function

(i) If a function is the product and quotient of functions such as y = f1(x) f2(x) f3(x) / g1(x) g2(x) g3(x) , we first take algorithm and then differentiate.
(ii) If a function is in the form of exponent of a function over another function such as [f(x)]g(x) , we first take logarithm and then differentiate.

#### Differentiation of a Function with Respect to Another Function

Let y = f(x) and z = g(x), then the differentiation of y with respect to z is dy / dz = dy / dx / dz / dx = f (x) / g (x)

#### Successive Differentiations

If the function y = f(x) be differentiated with respect to x, then the result dy / dx or f (x), so obtained is a function of x (may be a constant).
Hence, dy / dx can again be differentiated with respect of x.
The differential coefficient of dy / dx with respect to x is written as d /dx (dy / dx) = d2y / dx2 or f (x). Again, the differential coefficient of d2y / dx2 with respect to x is written as d / dx (d2y / dx2) = d3y / dx3 or f'(x)
Here, dy / dx, d2y / dx2, d3y / dx3, are respectively known as first, second, third,  order differential coefficients of y with respect to x. These alternatively denoted by f (x), f (x), f
(x),  or y1, y2, y3., respectively.
Note dy / dx = (dy / d?) / (dx / d?) but d2y / dx2 ? (d2y / d?2) / (d2x / d?2)

#### Leibnitz Theorem

If u and v are functions of x such that their nth derivative exist, then

#### Derivatives of Special Types of Functions

##### (viii) Differentiation of Integrable Functions

If g1 (x) and g2 (x) are defined in [a, b], Differentiable at x ? [a, b] and f(t) is continuous for g1(a) ? f(t) ? g2(b), then

##### Partial Differentiation

The partial differential coefficient of f(x, y) with respect to x is the ordinary differential coefficient of f(x, y) when y is regarded as a constant. It is a written as ?f / ?x or Dxf or fx.

e.g., If z = f(x, y) = x4 + y4 + 3xy2 + x4y + x + 2y
Then, ?z / ?x or ?f / ?x or fx = 4x3 + 3y2 + 2xy + 1 (here, y is consider as constant) ?z / ?y or ?f / ?y or fy = 4y3 + 6xy + x2 + 2 (here, x is consider as constant)

##### Higher Partial Derivatives

Let f(x, y) be a function of two variables such that ?f / ?x , ?f / ?y both exist.

(i) The partial derivative of ?f / ?y w.r.t. x is denoted by ?2f / ?x2 / or fxx.
(ii) The partial derivative of ?f / ?y w.r.t. y is denoted by ?2f / ?y2 / or fyy.
(iii) The partial derivative of ?f / ?x w.r.t. y is denoted by ?2f / ?y ?x / or fxy.
(iv) The partial derivative of ?f / ?x w.r.t. x is denoted by ?2f / ?y ?x / or fyx.

Note ?2f / ?x ?y = ?2f / ?y ?x
These four are second order partial derivatives.

##### Eulers Theorem on Homogeneous Function

If f(x, y) be a homogeneous function in x, y of degree n, then x (&partf / ?x) + y (&partf / ?y) = nf

##### Deduction Form of Eulers Theorem

If f(x, y) is a homogeneous function in x, y of degree n, then
(i) x (?2f / ?x2) + y (?2f / ?x ?y) = (n  1) &partf / ?x
(ii) x (?2f / ?y ?x) + y (?2f / ?y2) = (n  1) &partf / ?y
(iii) x2 (?2f / ?x2) + 2xy (?2f / ?x ?y) + y2 (?2f / ?y2) = n(n  1) f(x, y)

##### Important Points to be Remembered

If ? is m times repeated root of the equation f(x) = 0, then f(x) can be written as f(x) =(x  ?)m g(x), where g(?) ? 0.
From the above equation, we can see that f(?) = 0, f (?) = 0, f (?) = 0,  , f(m  l) ,(?) = 0.
Hence, we have the following proposition f(?) = 0, f (?) = 0, f (?) = 0,  , f(m  l) ,(?) = 0.
Therefore, ? is m times repeated root of the equation f(x) = 0.

PART 2

math notes for Class 12 Download PDF Continuity and Differentiability Chapter 5
math notes for Class 12 Download PDF Continuity and Differentiability Chapter 5