Lecture-10

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 10
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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।