Some Important Results of Derivatives:

\frac{d}{dx} (x) = 1
\frac{d}{dx} (constant) = 0
\frac{d}{dx} (x^n) = n{x}^{n-1}
\frac{d}{dx} (\log x) = \frac{1}{x}
\frac{d}{dx} (e^x) = e^x
\frac{d}{dx} (a^x) = {a^x} \log a
\frac{d}{dx} (\sqrt{x}) = \frac{1}{2\sqrt{x}}

\frac{d}{dx} (\sin x) = \cos x
\frac{d}{dx} (\cos x) = – \sin x
\frac{d}{dx} (\tan x) = \sec^{2} x
\frac{d}{dx} (\sec x) = \sec x . \tan x
\frac{d}{dx} (\cosec x) = – \cosec x . \cot x
\frac{d}{dx} (\cot x) = – \cosec^2 x

\frac{d}{dx} (sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}
\frac{d}{dx} (cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx} (tan^{-1}x) = \frac{1}{1+x^2}
\frac{d}{dx} (cot^{-1}x) = -\frac{1}{1+x^2}
\frac{d}{dx} (sec^{-1}x) = \frac{1}{x{\sqrt{x^2-1}}}
\frac{d}{dx} (cosec^{-1}x) = -\frac{1}{x{\sqrt{x^2-1}}}

PRODUCT RULE:
(I \times II)’ = I (II)’ + II (I)’

QUOTIENT RULE:
\left( \frac{N}{D} \right )^’ = \frac{D(N)’ – N(D)’}{D^2}

LOGARITHM RULES:
\log (m \times n) = \log m + \log n
  \log \left ( \frac{m}{n} \right ) = \log m – \log n
\log (m)^n = n \log m
\log_a (b) = \frac{\log_x (b)}{\log_x (a)}
\log_e(e^x) = x
\log_a(a^x) = x
(e)^{\log_e{x}} = x
(a)^{\log_a{x}} = x

Continuity and Differentiability Lecture 12
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This lecture is the continuation of the previous lecture based on miscellaneous questions from NCERT Miscellaneous Exercise of Chapter 5 Class 12 Maths Continuity and Differentiability.

Questions discussed in this lecture:

NCERT MISCELLANEOUS EXERCISE

Question 12. Find \frac{dy}{dx} , if y = 12(1 – \cos t), x = 10(t – \sin t), \frac{-\pi}{2}<t<\frac{\pi}{2}

Question 13. Find \frac{dy}{dx} , if y = \sin^{-1}x + \sin^{-1}{\sqrt{1 – x^2}}, 0<x<1

Question 14. If x\sqrt{1 + y} + y\sqrt{1 + x} = 0, for, -1<x<1 , prove that \frac{dy}{dx} = – \frac{1}{(1 + x)^2}

Question 15. If (x – a)^2 + (y – b)^2 = c^2 , for some c > 0, prove that \frac{\left[ 1 + \left( \frac{dy}{dx}\right )^2 \right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} is a constant independent of a and b.

Question 16. If \cos y = x \cos (a + y) , with \cos a \ne \pm 1 , prove that \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a} .

Question 17. If x = a (\cos t + t \sin t) \rm{and} y = a (\sin t – t \cos t), find \frac{d^2{x}}{dx^2} .

Question 18. If f(x) = | x |^3 , show that f ”(x) exists for all real x and find it.

Question 19. Using mathematical induction prove that \frac{d}{dx}(x^n) = nx^{n – 1} for all positive integers n.

Question 20. Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Question 21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

Question 22. If \begin{vmatrix}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \\ \end{vmatrix} prove that \frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\  l & m & n \\ a & b & c \\ \end{vmatrix}​

Question 23. If y = e^{a \cos^{-1}x}, -1 \le x \le 1 , show that (1 – x^2) \frac{d^2{y}}{dx^2} – x \frac{dy}{dx} – a^2 y = 0

राह संघर्ष की जो चलता है,
वो ही संसार को बदलता है,
जिसने रातों से जंग जीती है,
सूर्य बन कर वो ही निकलता है।