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 9
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In this lecture, I am discussing about parametric functions and questions from NCERT Exercise 5.6 which are based on differentiation of parametric functions.

Questions discussed in this lecture:

NCERT EXERCISE 5.6 (Parametric Functions)

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find \frac{dy}{dx} .

Question 1. x = 2at^2, y = at^4

Question 2. x = a \cos \theta, y = b \cos \theta

Question 3. x = \sin t, y = \cos 2t

Question 4.   x = 4t, y = \frac{4}{t}

Question 5. x = \cos \theta – \cos 2\theta, y = \sin \theta – \sin 2\theta

Question 6. x = a( \theta – \sin \theta), y a (1 + \cos \theta)

Question 7. x = \frac{sin^3{t}}{\sqrt{\cos 2t}},  y = \frac{cos^3{t}}{\sqrt{\cos 2t}}

Question 8. x = a \left( \cos t + \log \tan \frac{t}{2} \right ), y = a \sin t

Question 9. x = a \sec \theta, y = b \tan \theta

Question 10. x = a(\cos \theta + \theta \sin \theta), y = a(\sin \theta – \theta \cos \theta)

Question 11. If x = \sqrt{a^{sin^{-1}t}},  y = \sqrt{a^{cos^{-1}t}}, show that \frac{dy}{dx} = -\frac{y}{x} .

सफर में मुश्किलें आऐ, तो हिम्मत और बढ़ती है।
कोई अगर रास्ता रोके, तो जुर्रत और बढ़ती है।
अगर बिकने पे आ जाओ, तो घट जाते हैं दाम अक्सर।
ना बिकने का इरादा हो तो, कीमत और बढ़ती है।