Math Notes For Class 12 Download PDF Continuity And Differentiability Chapter 5

Chapter 5: Continuity and Differentiability

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 .

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).

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

1. d / d(x) (x^{n}) = nx^{n – 1}, x ∈ R, n ∈ R

2. d / d(x) (k) = 0, where k is constant.

3. d / d(x) (e^{x}) = e^{x}

4. d / d(x) (a^{x}) = a^{x} loge a > 0, a ≠ 1

(v) if d / d(x) f(x) = φ(x), then d / d(x) f(ax + b) = a φ(ax + b)

(vi) Differentiation of a constant function is zero i.e., d / d(x) (c) = 0.

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).

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.

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

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.

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)

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

(i) If a function is the product and quotient of functions such as y = f_{1}(x) f_{2}(x) f_{3}(x)… / g_{1}(x)
g_{2}(x) g_{3}(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.

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)

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) = d^{2}y / dx^{2} or f’ (x). Again, the differential coefficient of d^{2}y / dx^{2} with respect to x is written as d / dx (d^{2}y / dx^{2}) = d^{3}y / dx^{3} or f”'(x)……

Here, dy / dx, d^{2}y / dx^{2}, d^{3}y / dx^{3},… 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 y_{1}, y_{2}, y_{3}…., respectively.

Note dy / dx = (dy / dθ) / (dx / dθ) but d^{2}y / dx^{2} ≠ (d^{2}y / dθ^{2}) / (d^{2}x / dθ^{2})

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

If g_{1} (x) and g_{2 }(x) are defined in [a, b], Differentiable at x ∈ [a, b] and f(t) is continuous for g_{1}(a) ≤ f(t) ≤ g_{2}(b), then

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 D_{x}f or f_{x}.

e.g., If z = f(x, y) = x_{4} + y^{4} + 3xy^{2} + x^{4}y + x + 2y

Then, ∂z / ∂x or ∂f / ∂x or f_{x} = 4x^{3} + 3y^{2} + 2xy + 1 (here, y is consider as constant) ∂z / ∂y or ∂f / ∂y or fy = 4y^{3} + 6xy + x^{2} + 2 (here, x is consider as constant)

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 ∂^{2}f / ∂x^{2} / or fxx.

(ii) The partial derivative of ∂f / ∂y w.r.t. ‘y’ is denoted by ∂^{2}f / ∂y^{2} / or fyy.

(iii) The partial derivative of ∂f / ∂x w.r.t. ‘y’ is denoted by ∂^{2}f / ∂y ∂x / or fxy.

(iv) The partial derivative of ∂f / ∂x w.r.t. ‘x’ is denoted by ∂^{2}f / ∂y ∂x / or fyx.

Note ∂^{2}f / ∂x ∂y = ∂^{2}f / ∂y ∂x

These four are second order partial derivatives.

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

If f(x, y) is a homogeneous function in x, y of degree n, then

(i) x (∂^{2}f / ∂x^{2}) + y (∂^{2}f / ∂x ∂y) = (n – 1) &partf / ∂x

(ii) x (∂^{2}f / ∂y ∂x) + y (∂^{2}f / ∂y^{2}) = (n – 1) &partf / ∂y

(iii) x^{2} (∂^{2}f / ∂x^{2}) + 2xy (∂^{2}f / ∂x ∂y) + y^{2} (∂^{2}f / ∂y^{2}) = n(n – 1) f(x, y)

If α is m times repeated root of the equation f(x) = 0, then f(x) can be written as f(x) =(x – α)^{m} g(x), where g(α) ≠ 0.

From the above equation, we can see that f(α) = 0, f’ (α) = 0, f” (α) = 0, … , f^{(m – l)} ,(α) = 0.

Hence, we have the following proposition f(α) = 0, f’ (α) = 0, f” (α) = 0, … , f^{(m – l)} ,(α) = 0.

Therefore, α is m times repeated root of the equation f(x) = 0.

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- Chapter 5: Continuity and Differentiability
- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 6: Application of Derivatives
- Chapter 7: Integrals
- Chapter 8: Application of Integrals
- Chapter 9. Differential Equations
- Chapter 10: Vector Algebra
- Chapter 12: Linear Programming
- Chapter 11: Three Dimensional Geometry

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