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:
$$\left( \frac{N}{D} \right )^’ = \frac{D(N)’ – N(D)’}{D^2}$$

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In this lecture, I am discussing few questions based on Chain Rule, introduction to implicit functions, difference between implicit and explicit functions, questions from NCERT Exercise 5.2, NCERT Exercise 5.3 and Example 26, 27.

Questions discussed in this lecture:

## NCERT EXERCISE 5.2 (Chain Rule)

Differentiate the functions wih respect to x in Exercises 1 to 8:
Question 1. $\sin(x^2 + 5)$

Question 2. $\cos (\sin x)$

Question 3. $\sin (ax + b)$

Question 4. $\sec (\tan ( \sqrt{x}))$

Question 5. $\frac{\sin (ax+b)}{\cos (cx+d)}$

Question 6. $\cos {x^3}. \sin^2 (x^5)$

Question 7. $2 \sqrt{\cot({x^2})}$

Question 8. $\cos (\sqrt{x})$

## NCERT EXAMPLES

Example 26 Find the derivative of f given by $f(x) = sin^{–1} x$ assuming it exists.

Example 27 Find the derivative of f given by $f(x) = tan^{–1} x$ assuming it exists.

## NCERT EXERCISE 5.3 (Implicit & Explicit Functions)

Find $\frac{dy}{dx}$ in the following:
Question 1. $2x + 3y = \sin x$

Question 2. $2x + 3y = \sin y$

Question 3. $ax + by^2 = \cos y$

Question 4. $xy + y^2 = \tan x + y$

Question 5. $x^2 + xy + y^2 =100$

Question 6. $x^3 + x^2{y} + x{y^2} + y^3 = 81$

Question 7. $\sin^2 y + \cos xy = \kappa$

Question 8. $\sin^2 y + \cos^2 y = 1$

Question 9. $y = \sin^{-1} \left( \frac{2x}{1+x^2} \right)$

Question 10. $y = \tan^{-1} \left( \frac{3x-{x^3}}{1-3{x^2}} \right), -\frac{1}{\sqrt{3}}

Question 11. $y = \cos^{-1} \left( \frac{1-x^2}{1+x^2} \right), 0

​​​​​​Question 12. $y = \sin^{-1} \left( \frac{1-x^2}{1+x^2} \right), 0​​​​​​​

Question 13. $y = \cos^{-1} \left( \frac{2x}{1+x^2} \right), -1

​Question 14. $y = \sin^{-1} \left( 2x \sqrt{1-x^2} \right), -\frac{1}{\sqrt{2}}

​Question 15. $y = \sec^{-1} \left( \frac{1}{2{x^2}-1} \right), 0

टूटने लगे हौसले तो ये याद रखना,
बिना मेहनत के तख्तो-ताज नहीं मिलते,
ढूंढ़ लेते हैं अंधेरों में मंजिल अपनी,
क्योंकि जुगनू कभी रौशनी के मोहताज़ नहीं होते |