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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In this lecture, I am discussing about second order differentiation, which is also known as double derivatives.Also, I am discussing questions from NCERT Exercise 5.7 which are based on second order derivatives.

Questions discussed in this lecture:

## NCERT EXERCISE 5.7 (Second Order Derivatives)

Find the second order derivatives of the functions given in Exercises 1 to 10.

Question 1. $x^2 + 3x + 2$

Question 2. $x^20$

Question 3. $x. \cos x$

Question 4. $\log x$

Question 5. $x^3 \log x$

Question 6. $e^x \sin 5x$

Question 7. $e^{6x} \cos 3x$

Question 8. $\tan^{-1}x$

Question 9. $\log(\log x)$

Question 10. $\sin(\log x)$

Question 11. If $y = 5 \cos x – 3 \sin x$, prove that $\frac{d^2{y}}{dx^2} + y = 0$.

Question 12. If $y = \cos^{-1}x$, Find $\frac{d^2{y}}{dx^2}$ in terms of y alone.

Question 13. If $y = 3 \cos(\log x) + 4\sin(\log x)$, show that $x^2{y_2} + x{y_1} + y = 0$.

Question 14. If $y = Ae^{mx} + Be^{nx}$, show that $\frac{d^2{y}}{dx^2} – (m + n)\frac{dy}{dx} + mny = 0$.

Question 15. If $y = 500e^{7x} + 600e^{-7x}$, show that $\frac{d^2{y}}{dx^2} = 49y$.

Question 16. If $e^y(x+1) = 1$, show that $\frac{d^2{y}}{dx^2} = \left( \frac{dy}{dx} \right)^2$.

Question 17. If $y = (tan^{-1}x)^2$, show that $(x^2 + 1)^2 y_2 + 2x (x^2 + 1) y_1 = 2$.

“ख्वाहिशों” से नही गिरते हैं, “फूल” झोली में,
कर्म की साख़ को हिलाना होगा,
कुछ नही होगा कोसने से किस्मत को,
अपने हिस्से का दीया ख़ुद ही जलाना होगा।