JEE Mathematics Continuity and Differentiability

Method Of Differentiation Formulas

Differentiation of some elementary functions:
1. \( \frac{d}{dx} (x^n) = nx^{n-1} \)
2. \( \frac{d}{dx} (a^x) = a^x \ln a \)
3. \( \frac{d}{dx} (\ln |x|) = \frac{1}{x} \)
4. \( \frac{d}{dx} (\log_a x) = \frac{1}{x \ln a} \)
5. \( \frac{d}{dx} (\sin x) = \cos x \)
6. \( \frac{d}{dx} (\cos x) = -\sin x \)
7. \( \frac{d}{dx} (\sec x) = \sec x \tan x \)
8. \( \frac{d}{dx} (\text{cosec } x) = -\text{cosec } x \cot x \)
9. \( \frac{d}{dx} (\tan x) = \sec^2 x \)
10. \( \frac{d}{dx} (\cot x) = -\text{cosec}^2 x \)

Basic Theorems:
1. \( \frac{d}{dx} (f \pm g) = f'(x) \pm g'(x) \)
2. \( \frac{d}{dx} (k f(x)) = k \frac{d}{dx} f(x) \)
3. \( \frac{d}{dx} (f(x) \cdot g(x)) = f(x) g'(x) + g(x) f'(x) \)
4. \( \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f'(x) - f(x) g'(x)}{g^2(x)} \)
5. \( \frac{d}{dx} (f(g(x))) = f'(g(x)) g'(x) \)

Derivative Of Inverse Trigonometric Functions:
\( \frac{d \sin^{-1} x}{dx} = \frac{1}{\sqrt{1-x^2}}, \frac{d \cos^{-1} x}{dx} = -\frac{1}{\sqrt{1-x^2}}, \text{ for } -1 < x < 1. \)
\( \frac{d \tan^{-1} x}{dx} = \frac{1}{1+x^2}, \frac{d \cot^{-1} x}{dx} = -\frac{1}{1+x^2}, (x \in R) \)
\( \frac{d \sec^{-1} x}{dx} = \frac{1}{|x|\sqrt{x^2-1}}, \frac{d \text{cosec}^{-1} x}{dx} = -\frac{1}{|x|\sqrt{x^2-1}}, \text{ for } x \in (-\infty, -1) \cup (1, \infty) \)

Differentiation using substitution:
(i) \( \sqrt{x^2 + a^2} \) by substituting \( x = a \tan \theta, \text{ where } -\frac{\pi}{2} < \theta < \frac{\pi}{2} \)
(ii) \( \sqrt{a^2 - x^2} \) by substituting \( x = a \sin \theta, \text{ where } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2} \)
(iii) \( \sqrt{x^2 - a^2} \) by substituting \( x = a \sec \theta, \text{ where } \theta \in [0, \pi], \theta \neq \frac{\pi}{2} \)
(iv) \( \sqrt{\frac{x+a}{a-x}} \) by substituting \( x = a \cos \theta, \text{ where } \theta \in (0, \pi] \).

Parametric Differentiation:
If \( y = f(\theta) \) & \( x = g(\theta) \) where \( \theta \) is a parameter, then \( \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \).

Derivative of one function with respect to another:
Let \( y = f(x); z = g(x) \) then \( \frac{dy}{dz} = \frac{dy/dx}{dz/dx} = \frac{f'(x)}{g'(x)} \).