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 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।