By going through these CBSE Class 12 Maths Notes Chapter 5 Continuity and Differentiability, students can recall all the concepts quickly.

## Continuity and Differentiability Notes Class 12 Maths Chapter 5

**Continuity (Definition):** if f be a real-valued function on a subset of real numbers and let c be a point in its domain, then f is a continuous function at e, if

Obviously, if the left-hand limit and right-hand limit and value of the function at x = c exist and are equal to each other, i.e., if

then f is continuous at x = c.

**Algebra of continuous functions:**

Let f and g be two real functions, continuous at x = c, then

- Sum of two functions is continuous at x = c, i.e., (f + g) (x), defined as f(x) + g(x), is continuous at x = c.
- Difference of two functions is continuous at x = c, i.e., (f – g) (x), defined as f(x) – g(x), is continuous at x = c.
- Product of two functions is continuous at x = c, i.e., (f g) (x), defined as f(x) . g(x) is continuous at x = c.
- Quotient of two functions is continuous at x = c, (provided it is defined at x = c), i.e.,

(\(\frac{f}{g}\))(x), defined as \(\frac{f(x)}{g(x)}\) [g(x) ≠ 0], is continuous at x = c.

However, if f(x) = λ, then

(a) λ.g, defined ty .g(x), is also continuous at x = c.

(b) Similr1y, if \(\frac{λ}{g}\) is defined as \(\frac{λ}{g}\) (x) = \(\frac{λ}{g(x)}\) , then \(\frac{λ}{g}\) is also continuous at x = c.

→ Differentiability: The concept of differentiability has been introduced in the lower class. Let us recall some important results.

→ Differentiability (Definition): Let f be a real function and c is a point in its domain. The derivative of f at c is defined as

Every differentiable function is continuous.

→ Algebra of Derivatives: Let u and v be two functions of x.

- (u ± v)’ = u’ ± v’
- (uv)’ = u’v + uv’
- \(\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}\), where v ≠ 0.

→ Derivative of Composite Function: Let t be a real valued function which is a composite of two functions u and v, i.e., f = vou. Put u(x) = t and f= v(t).

∴ \(\frac{d f}{d x}=\frac{d v}{d t} \cdot \frac{d t}{d x}\)

→ Chain Rule: Let/be a real valued function which is a composite fimction of u, v and w, i.e., f(wov)ou.

Put u(x) = t, v(t) = s and f = w(s). Then,

\(\frac{d f}{d x}=\frac{d w}{d s} \cdot \frac{d s}{d t} \cdot \frac{d t}{d x}\).

→ Derivatives of Inverse Trigonometric Functions:

Functions |
Domain |
Derivatives |

Sin^{-1}x |
[- 1, 1] | \( \frac{1}{\sqrt{1-x^{2}}} \) |

Cos^{-1}x |
[- 1, 1] | \( -\frac{1}{\sqrt{1-x^{2}}} \) |

tan^{-1}x |
R | \( \frac{1}{1+x^{2}} \) |

Cot^{-1}x |
R | \( -\frac{1}{1+x^{2}} \) |

Sec^{-1}x |
(-∞, – 1] ∪ [1, ∞) | \( \frac{1}{x \sqrt{x^{2}-1}} \) |

Cosec^{-1}x |
(-∞, – 1] ∪ [1, ∞) | \( -\frac{1}{x \sqrt{x^{2}-1}} \) |

Implicit Functions: An equation in form f(x, y) = 0, in which y is not expressible in terms of x, is called an implicit function of x and y.

Both sides of the equations are differentiated termwise. Then, from this equation, \(\frac{d y}{d x}\) is obtained. It may be noted that when a function of y occurs, then differentiate it w.r.t. y and multiply it by \(\frac{d y}{d x}\).

e.g., To find \(\frac{d y}{d x}\) from cos^{2} y + sin xy = 1, we differentiate it as

**Exponential Functions:**

The exponential function, with positive base b > 1, is the function y = b^{x}.

- The graph of y = 10
^{x}is shown in the figure. - Domain = R
- Range = R
^{+} - The point (0,1) always lies on the graph.
- It is an increasing function, i.e., as we move from left to right, the graph rises above.
- As x → – ∞, y → 0.
- \(\frac{d}{dx}\) (a
^{x}) = a^{x}log, a, \(\frac{d}{dx}\) e^{x}= e^{x}.

**Logarithmic Functions:**

Let b> 1 be a real number. b^{x} = a may be written as log_{b} a = x.

- The graph of y = log
_{10}x is shown in the figure. - Domain = R
^{+}, Range = R. - It is an increasing function.
- As x → 0, y → ∞.
- The function y = e
^{x}and y = log_{e}x are the mirror images of each other in the line y = x. - \(\frac{d}{dx}\) (loga x) = \(\frac{1}{x}\) l0ga e, \(\frac{d}{dx}\) loge x = \(\frac{1}{x}\)

→ Other properties of Logarithm are:

- log
_{b}pq = log_{b}p + log_{b}q - log
_{b}\(\frac{p}{q}\) = log_{a}p – log_{a}q - log
_{b}px = x log_{b}p – log_{b}q - log
_{a}b = \(\frac{\log _{a} p}{\log _{b} p}\)

→ Logarithmic Differentiation:

Whenever the functions are given in the form

- y = [u(x)]v(x) and
- y = \(\frac{u(x) \times v(x)}{w(x)}\)

take log of both sides. Simplify and differentiate, e.g.,

Let y = (cos x)sin x, log y = sin x log cos x

Differentiating, \(\frac{1}{y}\) \(\frac{dy}{dx}\) = cos x log Cos x + sin x . – \(\frac{sin x}{cos x}\)

∴ \(\frac{dy}{dx}\) = (cos x)^{sin y} [cos x log cosx – sin x tan x].

→ Derivatives of Functions in Parametric Form: Let the given equations be x = f(t) and y = g(t), where t is the parameter. Then,

→ Second Order Derivative:

Let y = f(x), then \(\frac{dy}{dx}\) =f ‘(x).

If f ‘(x) is differentiable, then it is again differentiated.

**Rolle’s Theorem:**

Let f: [a, b] → R be continuous on closed interval [a, b] and differentiable on open interval (a, b) such that f(a) = f(b), where a and b are real numbers, then there exists some c ∈ (a, b) such that f ‘(c) = 0.

From the figure, we observe that f(a) = f(b). There exists a point c_{1} ∈ (a, b) such that f ‘ (c) = 0, i.e., tangent at c1 is parallel to x-axis. Similarly, f(b) = f(c) → f ‘ (c_{2}) = 0.

→ Mean Value Theorem: Let f: [a, b] → R be a continuous function on the closed interval [a, b] and differentiable in the open interval (a, b). Then, there exists some c ∈ (a, b) such that

f ‘ (c) = \(\frac{f(b)-f(a)}{b-a}\)

Now, we know that \(\frac{f(b)-f(a)}{b-a}\) is the slope of secant drawn between A[a,f(a)] and B[b,f(b)]. We t k know that the slope of the line joining (x_{1,} y_{1}) and (x_{2}, y_{2}) is \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

The theorem states that there is a point c ∈ (a, b), where f ‘(c) is equal to the slope of AB.

In other words, there exists a point c ∈ (a, b) such that tangent at x = c is parallel to AB.

1. CONTINUITY

(i) Left Continuity. A function ‘f ’ is left-continuous at x = c if \(\lim _{x \rightarrow c^{-}}\) f (x) = f(c).

(ii) Right Continuity. A function ‘f ’ is right-continuous at x = c if \(\lim _{x \rightarrow c^{+}}\) f (x) = f(c).

(iii) Continuity at a point. A function ‘ f ’ is continuous at x = c if

\(\lim _{x \rightarrow c^{-}}\) (x) = \(\lim _{x \rightarrow c^{+}}\) f(x) = f(c).

2. (i) Polynominal functions

(ii) Rational functions

(iii) Exponential functions

(iv) Trigonometric functions are all continuous at each point of their respective domain.

3. DIFFERENTIABILITY

(i) Left Derivative. A function ‘f ’ is said to possess left derivative at x = c if \(\lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{-h}\) exists finitely.

(ii) Right Derivative. A function ‘f ’ is said to possess right derivative at x = c if

\(\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}\) exists finitely.

(iii) Derivative. A function is said to possess derivative at x = c if \(\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}\) exists finitely.

4. CONTINUITY AND DERIVABILITY

A real valued function is finitely derivable at any point of its domain, it is necessarily continuous at that point. The converse is not true.

5. STANDARD RESULTS

(i) \(\frac{d}{d x}\) (x^{n}) = nx^{n-1} ∀ x ∈ R

(ii) \(\frac{d}{d x}\) ((ax + b)n = n(ax + b)^{n – 1} . a ∀ x ∈ R

(iii) \(\frac{d}{d x}(|x|)=\frac{x}{|x|}\), x ≠ 0

6. GENERAL THEOREMS

(i) The derivative of a constant is zero.

(ii) An additive constant vanishes on differentiation i.e. if f(x) = g(x) + c, where ‘c’ is any constant, then f'(x) = g'(x).

(iii) If f(x) = ag(x), then f'(x) = ag'(x), where ‘a’ is a scalar.

(iv) If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x).

Extension.

If f(x) = a_{1}f_{1} ± a_{2}f_{2} ……. ± a_{n}f_{n}(x), then :

f'(x) = a_{1}f_{1}‘(x) ± a_{2}f_{2}‘(x) ± ……. ± a_{n}f_{n}‘(x)

(v) If f(x) = \(\frac{g(x)}{h(x)}\), then f'(x) = g(x)h'(x) + g'(x)h(x)

(vi) If f(x) = \(\frac{g(x)}{h(x)}\), then f ‘(x) = \(\frac{h(x) g^{\prime}(x)-g(x) h^{\prime}(x)}{(h(x))^{2}}\), h(x) ≠ 0.

(vii) If f(x) = \(\frac{1}{h(x)}\), then f'(x) = \(-\frac{h(x)}{[h(x)]^{2}}\), h'(x) ≠ 0

7. IMPORTANT RESULTS

(i) (a) \(\frac{d}{d x}\) (sinx) = cos x and \(\frac{d}{d x}\) (cos x) = – sin x ∀ x ∈ R

(b) \(\frac{d}{d x}\) (tan x) = sec^{2} x and \(\frac{d}{d x}\) (sec x) = sec x tan x ∀ x ∈ R except odd multiples of \(\frac{\pi}{2}\)

(c) \(\frac{d}{d x}\)(cot x) = – cosec^{2} x and \(\frac{d}{d x}\) (cosec x) = -cosec x cot x ∀ x ∈ R except even multiple of \(\frac{\pi}{2}\)

(ii)

(a) \(\frac{d}{d x}\)(sin^{-1}x) = \(\frac{1}{\sqrt{1-x^{2}}}\), |x| < 1

(b) \(\frac{d}{d x}\)(cos^{-1}x) = \(-\frac{1}{\sqrt{1-x^{2}}}\), |x| < 1

(c) \(\frac{d}{d x}\)(tan^{-1}x) = \(\frac{1}{1+x^{2}}\) ∀ x ∈ R

(d) \(\frac{d}{d x}\)(cot^{-1}x) = \(-\frac{1}{1+x^{2}}\) ∀ x ∈ R

(e) \(\frac{d}{d x}\)(sec^{-1}x) = \(\frac{1}{|x| \sqrt{x^{2}-1}}\) x > 1 or x < -1

(f) \(\frac{d}{d x}\)cosec^{-1}x) = \(-\frac{1}{|x| \sqrt{x^{2}-1}}\), x > 1 or x < -1

(iii) (a) \(\frac{d}{d x}\) (a^{x}) = a^{x} log_{e} a, a > 0

(b) \(\frac{d}{d x}\)(e^{x}) = e^{x}

(c) \(\frac{d}{d x}\)(log_{a} x) = \(\frac{1}{x}\) log_{a} e, x > 0

(d) \(\frac{d}{d x}\) (log x) = \(\frac{1}{x}\), x>0.

8. CHAIN RULE

\(\frac{d}{d x}\) (f(g(x)) = f'(g(x)).g'(x)

9. PARAMETRIC EQUATIONS

\(\frac{d y}{d x}=\frac{d y / d t}{d x / d t}\), \(\frac{d x}{d t}\) ≠ 0

Or \(\frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}\)

10.MORE RESULTS

(i) \(\frac{d y}{d x}=\frac{1}{\frac{d x}{d y}}\)

(ii) \(\frac{d y}{d x} \times \frac{d x}{d y}=1\)

11. ROLLE’S THEOREM

If a function f(x) is :

(i) continuous in [a, b]

(ii) derivable in (a, b)

(iii) f (a) = f (b), then there exists at least one point ‘c’ in (a, b) such that f’ (c) = 0.

12. LAGRANGE’S MEAN VALUE THEROEM (LMV THEOREM OR MV THEOREM)

If a function f(x) is :

(i) continuous in [a, b]

(ii) derivable in (a, b), then there exists at least one point ‘c’ in (a, b) such that \(\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)\)