# L10 Continuity and Differentiability

“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।

# Lecture - 10 Chapter 5 Continuity and Differentiability

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 $(x^2 + 1)^2 y_2 + 2x (x^2 + 1) y_1 = 2$.