Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Find the derivatives of the following:

Question 1.
y = xcos x
Answer:
y = xcos x
Taking log on both sides
log y = log xcos x
log y = cos x log x
Differentiating with respect to x
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 1

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 2.
y = xlog x + (log x)x
Answer:
y = xlog x + (log x)x
Let u = xlog x, v = (log x)x
log u = log xlog x
log u = (log x) (log x)
log u = (log x)2
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 2
v = (log x)x
log v = log (log x)x
log v = x log (log x)
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 3

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 3.
\(\sqrt{x y}\) = e(x – y)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 4

Question 4.
xy = yx
Answer:
xy = yx
Taking log on both sides
log xy = log yx
y log x = x log y
Differentiate with respect to x
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 5

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 5.
(cos x)log x
Answer:
y = (cos x)log x
Taking log on both sides
log y = log (cos x)log x
log y = (log x) log (cos x)
Differentiating with respect to x
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 6

Question 6.
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 7

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 7.
\(\sqrt{x^{2}+y^{2}}=\tan ^{-1}\left(\frac{y}{x}\right)\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 8
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 9
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 10

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 8.
tan (x + y) + tan (x – y) = x
Answer:
tan (x + y) + tan (x – y) = x
Differentiating with respect to x
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 11

Question 9.
If cos(xy) = x, show that
\(\frac{d y}{d x}=\frac{-(1+y \sin (x y))}{x \sin x y}\)
Answer:
cos (xy) = x
Differentiating with respect to x
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 12

Question 10.
\(\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}\)
Answer:
Let y = \(\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}\)
[1 – cos 2θ = 2 sin2θ and 1 + cos 2θ = 2 sin2 θ]
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 13
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 14

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 11.
tan-1 = \(\left(\frac{6 x}{1-9 x^{2}}\right)\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 15
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 16

Question 12.
\(\cos \left(2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right)\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 17
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 18

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 13.
x = a cost ; y = a sin3t
Answer:
x = a cost , y = a sin3t
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 19

Question 14.
x = a (cos t + t sin t);
y = a (sin t – t cos t)
Answer:
x = a (cos t + t sin t) , y = a (sin t – t cos t)
\(\frac{d x}{d t}\) = a [- sin t + t cos t + sin t ]
\(\frac{d x}{d t}\) = at cos t —— (1)
y = a (sin t – t cos t)
\(\frac{d x}{d t}\) = a [cos t – (t × – sin t + cos t × 1)]
\(\frac{d x}{d t}\) = a[cos t + t sin t – cos t]
\(\frac{d x}{d t}\) = at sin t —— (2)
From equations (1) and (2) we get
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 20

Question 15.
x = \(\frac{1-t^{2}}{1+t^{2}}\) , y = \(\frac{2 t}{1+t^{2}}\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 21

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 16.
cos-1\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\)
Answer:
Let y = cos-1\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\)
Put x = tan θ
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 22
y = cos-1 (cos 2θ)
y = 2θ
y = 2 tan-1 x
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 23

Question 17.
sin-1 (3x – 4x3)
Answer:
Let y = sin-1 (3x – 4x3)
Put x = sin θ
y = sin-1 (3 sin θ – 4 sin3 θ)
y = sin-1 (sin 3θ)
y = 3θ
y = 3 sin-1 x
\(\frac{d y}{d x}=\frac{3}{\sqrt{1-x^{2}}}\)

Question 18.
tan-1 \(\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)\)
Answer:
Let y = tan-1 \(\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)\)
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 24

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 19.
Find the derivative of sin x2 with respect to x2.
Answer:
Let u = sin x2
\(\frac{\mathrm{d} \mathrm{u}}{\mathrm{d} x}\) = cos (x2) × 2x
\(\frac{\mathrm{d} \mathrm{u}}{\mathrm{d} x}\) = 2x cos (x2)
Let v = x2
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 25

Question 20.
Find the derivative of sin-1\(\left(\frac{2 x}{1+x^{2}}\right)\) with respect to tan-1 x.
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 26

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 21.
If u = tan-1\(\frac{\sqrt{1+x^{2}}-1}{x}\) and
v = tan-1x, find \(\frac{\mathrm{du}}{\mathrm{dv}}\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 27
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 28
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 29
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 30

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 22.
Find the derivative with tan-1\(\left(\frac{\sin x}{1+\cos x}\right)\) with respect to tan-1\(\left(\frac{\cos x}{1+\sin x}\right)\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 31
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 32
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 33

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 23.
If y = sin-1x then find y”.
Answer:
y = sin-1 x
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 34

Question 24.
If y = etan-1x, show that
(1 + x2) y” + (2x – 1) y’ = 0
Answer:
y = etan-1x
y’ = etan-1 × \(\frac{1}{1+x^{2}}\)
(1 + x2) y’ = etan-1x
(1 + x2) y’ = y
Differentiating with respect to x
(1 + x2) y” + y’ (0 + 2x) = y’
(1 + x2) y” + 2xy’ – y’ = 0
(1 + x2) y” + (2x – 1) y’ = 0

Question 25.
If y = \(\frac{\sin ^{-1} x}{\sqrt{1-x^{2}}}\), show that (1 – x2)y2 – 3xy1 – y = 0.
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 35
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 36

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 26.
If x = a (θ + sin θ), y = a (1 – cos θ) then prove that at θ = \(\frac{\pi}{2}\), y” = \(\frac{1}{a}\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 37
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 38

Question 27.
If sin y = x sin (a + y), the prove that \(\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}\), a ≠ nπ
Answer:
Given sin y = x sin (a + y) ——- (1)
Differentiating with respect to x , we get
cos y \(\frac{\mathrm{dy}}{\mathrm{d} x}\) = x cos (a + y) (0 + \(\frac{\mathrm{dy}}{\mathrm{d} x}\)) + sin (a + y) . 1
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 39

Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4

Question 28.
If y = (cos-1 x)2, prove that (1 – x2) \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} x^{2}}\) – x \(\frac{\mathrm{dy}}{\mathrm{d} x}\) – 2 = 0. Hence find y2 when x = 0.
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 40
Samacheer Kalvi 11th Maths Guide Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 41

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