AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b)

Practicing the Intermediate 1st Year Maths 1B Textbook Solutions Chapter 9 అవకలనం Exercise 9(b) will help students to clear their doubts quickly.

AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Exercise 9(b)

అభ్యాసం 9 (బి)

I.

ప్రశ్న 1.
కింది ప్రమేయాల అవకలజాలను కనుక్కోండి.

i) cotn x
సాధన:
f(x) = cotn x, \(\frac{d y}{d x}\) = n. cotn-1x.\(\frac{\mathrm{d}}{\mathrm{dx}}\)(cot x)
= n. cotn -1 x (- cosec2 x)
= – n. cotn -1x. cosec2 x

ii) cosec4 x
\(\frac{d y}{d x}\) = 4. cosec3 x. \(\frac{\mathrm{d}}{\mathrm{dx}}\) (cosec x)
= 4. cosec3 x (- cosec x. cot x)
= -4. cosec4 x. cot x

iii) tan (ex)
సాధన:
f(x) = tan (ex)
\(\frac{d y}{d x}\) = sec2 (ex). (ex)1 = ex. sec2 (ex)

AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b)

iv) \(\frac{1-\cos 2 x}{1+\cos 2 x}\)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 31

v)
sinmx. cosnx
సాధన:
f(x) = sinmx. cosnx
\(\frac{d y}{d x}\) = (sinmx). \(\frac{\mathrm{d}}{\mathrm{dx}}\)(cosnx)\(\frac{\mathrm{d}}{\mathrm{dx}}\)(sinmx)
= sinmxn cosn – 1x(-sin x) + cosnx. m sinm – 1x. cos x
= m. cosn + 1x. sinm – 1x – n. sinm + 1 x. cosn – 1x.

vi) sin mx. cos nx
సాధన:
f(x) = sin mx. cos nx
\(\frac{d y}{d x}\) = sin mx \(\frac{d}{d x}\)(cos nx) + (cos nx)\(\frac{d}{d x}\)(sin mx)
= sin mx (-n sin nx) + cos nx (m cos mx)
= m. cos mx. cos nx – n. sin mx. sin nx

vii) x tan-1 x
సాధన:
f(x) = x tan-1x\(\frac{d y}{d x}\)
= x.\(\frac{\mathrm{d}}{\mathrm{dx}}\) (tan-1x) + (tan-1 x)\(\frac{d}{d x}(x)\)
= \(\frac{x}{1+x^2}\) + tan-1x

viii) sin-1 (cos x)
సాధన:
f(x) = sin-1(cos x) = sin-1\(\left[\sin \left(\frac{\pi}{2}-x\right)\right]\) = \(\frac{\pi}{2}\) – x
\(\frac{d y}{d x}\) = 0 – 1 = -1

ix) log (tan 5x)
సాధన:
f(x) = log (tan 5x)
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 32

x) sinh-1\(\left(\frac{3 x}{4}\right)\)
సాధన:
f(x) = sinh-1\(\left(\frac{3 x}{4}\right)\)
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 33

xi) tan-1 (log x)
సాధన:
f(x) = tan-1(log x)
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{1+(\log x)^2}\).\(\frac{d}{d x}(\log x)\)
= \(\frac{1}{x\left(1+(\log x)^2\right)}\)

AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b)

xii) log \(\left(\frac{x^2+x+2}{x^2-x+2}\right)\) (May ’06)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 34

xiii) log (sin-1(ex))
సాధన:
f(x) = log(sin-1(ex))
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 35

xiv) (sin x)2(sin-1x)2
సాధన:
f(x) = (sin x)2(sin-1x)2
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 36

xv) \(\frac{\cos x}{\sin x+\cos x}\)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 37

xvi) \(\frac{x\left(1+x^2\right)}{\sqrt{1-x^2}}\)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 38

xvii) \(e^{\sin ^{-1} x}\)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 39

xviii) cos (log x + ex)
సాధన:
f(x) = cos (log x + ex)
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = – sin (log x + ex) \(\frac{d}{d x}\)(log x + ex)
= – sin (log x + ex)(\(\frac{1}{x}\) + ex)

xix) \(\frac{\sin (x+a)}{\cos x}\)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 40

AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b)

xx) cot-1 (cosec 3x)
సాధన:
f(x) = cot-1 (cosec 3x)
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 41

ప్రశ్న 2.
x దృష్ట్యా క్రింది వాటి అవకలజాలను కనుక్కోండి.

i) x = sinh2 y
సాధన:
f(x) = x = sinh2 y
\(\frac{d x}{d y}\) = 2 sinh y . cosh y
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 42

ii) x = tanh2/sup> y
సాధన:
f(x) = tanh2/sup> y. \(\frac{d x}{d y}\) = 2 tanh y. sech2 y
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 43

iii) x = esinh y
సాధన:
\(\frac{d x}{d y}\) = esinh y\(\frac{d}{d x}(\sinh y)\)
= esinh y. cosh y
= x . cosh y
\(\frac{d y}{d x}\) = \(\frac{1}{\left(\frac{d x}{d y}\right)}\) = \(\frac{1}{x \cdot \cosh y}\)

iv) x = tan (e-y)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 44

v) x = log (1 + sin2 y)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 45

vi) x = log (1 + \(\sqrt{\mathbf{y}}\))
సాధన:
1 + \(\sqrt{\mathbf{y}}\) = ex
\(\sqrt{\mathbf{y}}\) = ex – 1
y = (ex – 1)2
\(\frac{d y}{d x}\) = 2(ex – 1). ex = 2\(\sqrt{y}\). ex
= 2\(\sqrt{y}\) (\(\sqrt{y}\) + 1)
= 2(y + \(\sqrt{y}\))

II. కింది ప్రమేయాల అవకాలను కనుక్కోండి

i) y = cos (log (cot x))
సాధన:
y = cos (log (cot x))
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 46

AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b)

ii) sinh-1 \(\left(\frac{1-x}{1+x}\right)\)
సాధన:
y = sinh-1\(\left(\frac{1-x}{1+x}\right)\)
సందర్బ౦: 1. x < -1
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 47
సందర్బ౦: 2. x > -1
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 48

iii) log (cot(1 – x2))
సాధన:
y = log (cot(1 – x2))
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 49
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 50

iv) sin (cos (x2))
సాధన:
y = sin (cos (x2))
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = cos (cos (x2)). \(\frac{\mathrm{d}}{\mathrm{dx}}\) (cos (x2))
= cos (cos (x2)) (-sin (x2)). \(\frac{d}{d x}\)(x2)
= -2x. sin (x2). cos (cos (x2))

v) sin (tan-1 (ex))
సాధన:
y = sin (tan-1 (ex)
\(\frac{d y}{d x}\) = cos (tan-1(ex)). \(\frac{d}{d x}\)(tan-1(ex))
= cos(tan-1(ex)) . \(\left[\frac{1}{1+\left(e^x\right)^2}\right]\)(ex)
= \(\frac{e^x}{1+e^{2 x}}\). cos(tan-1(ex))

vi) \(\frac{\sin (a x+b)}{\cos (c x+d)}\)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 51

vii) tan-1\(\left[\tanh \left[\frac{x}{2}\right]\right]\)
సాధన:
y = tan-1 \(\left[\tanh \left[\frac{x}{2}\right]\right]\)
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 52

viii)
sin x. (Tan-1 x)2
సాధన:
y = sin x. (Tan-1 x)2
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 53

III. కింది ప్రమేయాల అవకలజాలను కనుక్కోండి.

ప్రశ్న 1.
sin-1 \(\left(\frac{\mathbf{b}+\mathbf{a} \sin x}{\mathbf{a}+\mathbf{b} \sin x}\right)\) (a > 0, b > 0)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 54
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 55

AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b)

ప్రశ్న 2.
cos-1\(\left(\frac{b+a \cos x}{a+b \cos x}\right)\) (a > 0, b > 0)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 56
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 57

ప్రశ్న 3.
Tan-1 \(\left[\frac{\cos x}{1+\cos x}\right]\)
సాధన:
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 58
AP Inter 1st Year Maths 1B Solutions Chapter 9 అవకలనం Ex 9(b) 59