## Lecture-11

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 miscellaneous questions of differentiation which are based on chain rule, logarithmic differentiation, derivative of implicit functions, derivative of parametric functions etc. All these questions are from NCERT Miscellaneous Exercise of Chapter 5 Class 12 Maths Continuity and Differentiability.

Questions discussed in this lecture:

## NCERT MISCELLANEOUS EXERCISE

Differentiate w.r.t. x the function in Exercises 1 to 11.

Question 1. $(3x^2 – 9x + 5)^9$

Question 2. $\sin^3{x} + \cos^6{x}$

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

Question 4. $\sin^{-1}(x\sqrt{x}), 0 \le x \le 1$

Question 5. $\frac{\cos^{-1}\frac{x}{2}}{\sqrt{2x+7}}, -2

Question 6. $\cot^{-1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 – \sin x}}{\sqrt{1 + \sin x} – \sqrt{1 – \sin x}} \right], 0

Question 7. $(\log x)^{\log x}, x>1$

Question 8. $\cos(a \cos x + b \sin x)$, for some constant a and b.

Question 9. $(\sin x – \cos x)^{(\sin x – \cos x)}, \frac{\pi}{4}

Question 10. $x^x + x^a + a^x + a^a$

Question 11. $x^{x^2 – 3} + (x – 3)^{x^2}$

रो कर मुस्कुराने का मजा ही कुछ और होता है,
जिंदगी में कुछ खो कर पाने का मजा ही कुछ और होता है,
ज़िन्दगी में हार और जीत तो लगी रहती है,
लेकिन हार के जीतने का मजा ही कुछ और होता है।