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

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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}

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

Question 14. If $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$, for, $-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।