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

Find the derivatives of the following functions
Question 1.
y = xcos x
Solution:
y = xcos x
Taking log on both sides
log y = log xcos x = cos x log x
differentiating w.r.to x we get
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 1

Question 2.
y = xlogx + (logx)x
Solution:
y = xlogx + (logx)x
Let y = u + v
Then \(\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}\)
u = xlogx
Taking log on both sides
log u = log x log x = log (x)2
differentiating w.r.to x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 2
Taking log on both sides
log u = log (logx)x = x log (log x)
differentiating w.r.to x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 3

Question 3.
\(\sqrt{x y}\) = e(x – y)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 4
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 5

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

Question 4.
xy = yx
Solution:
xy = yx
Taking log on both sides
logxy = logyx
(i.e.) y log x = x log y
differentiating w.r.to x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 6

Question 5.
(cos x)log x
Solution:
y = (cos x)log x
Taking log on both sides
log y = log (cos x)log x = log x (log cos x)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 7
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 8

Question 6.
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1
Solution:
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1
Differentiating w.r.to x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 9

Question 7.
\(\sqrt{x^{2}+y^{2}}=\tan ^{-1}\left(\frac{y}{x}\right)\)
Solution:
\(\sqrt{x^{2}+y^{2}}=\tan ^{-1}\left(\frac{y}{x}\right)\)
Differentiating w.r.to x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 10
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 11

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

Question 8.
tan (x + y) + tan (x – y) = x
Solution:
tan (x + y) + tan (x – y) = x
Differentiating w.r.to x we get
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 12

Question 9.
If cos (xy) = x, show that \(\frac{d y}{d x}=\frac{-(1+y \sin (x y))}{x \sin x y}\)
Solution:
cos (xy) = x
Differentiating w.r.to x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 13

Question 10.
\(\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 14

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

Question 12.
cos [2 \(\tan ^{-1} \sqrt{\frac{1-x}{1+x}}\)]
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 16

Question 13.
x = a cos3t; y = a sin2t
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 17

Question 14.
x = a (cos t + t sin t); y = a [sin t – t cos t]
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 18
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 19

Question 15.
x = \(\frac{1-t^{2}}{1+t^{2}}\); y = \(\frac{2 t}{1+t^{2}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 20

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

Question 16.
\(\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 21

Question 17.
sin-1 (3x – 4x3)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 22

Question 18.
\(\tan ^{-1}\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 23

Question 19.
Find the derivative of sin x2 with respect to x2
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 24

Question 20.
Find the derivative of \(\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\) with respect to tan-1 x.
Solution:
Let u = \(\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\) and v = tan-1 x
Now we have to find \(\frac{d u}{d v}\)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 25

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

Question 21.
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 26
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 27

Question 22.
Find the derivative with Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 28 with respect to Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 29
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 30
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 31

Question 23.
If y = sin-1 then find y”.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 32
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 33

Question 24.
If y = etan-1x, show that (1 + x2) y” + (2x – 1) y’ = 0
Solution:
y = etan-1x
y = etan-1x \(\left(\frac{1}{1+x^{2}}\right)\)
⇒ y’ = \(\frac{y}{1+x^{2}}\) ⇒ y'(1 + x2) = y
differentiating w.r.to x
y’ (2x) + (1 + x2) (y”) = y’
(i.e.) (1 + x2) y” + y’ (2x) – y’ = 0
(i.e.) (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
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 34
-xy + (1 – x2) y1 = 1
differentiating both sides again w.r.to x
-[x y1 + y (1)] + (1 – x2) (y2) + y1 (-2x) = 0
(i.e.) -xy1 – y + (1 – x2) y2 – 2xy1 = 0
(1 – x2) y2 – 3xy1 – y = 0

Question 26.
If x = a (θ + sin θ), y = a (1 – cos θ) then prove that at θ = \(\frac{\pi}{2}\), y” = \(\frac{1}{a}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 35
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 36

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

Question 27.
If sin y = x sin (a + y) Then prove that \(\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}\), a ≠ nπ
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 37

Question 28.
If y = (cos-1 x)2, prove that (1 – x2) \(\frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}\) – 2 = 0. Hence find y2 when x = 0.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 38

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

Question 1.
If y = A cos4x + B sin 4x, A and B are constants then Show that y2 + 16y = 0
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 39

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

Question 2.
If y = cos (m sin-1 x), prove that (1 – x2) y3 – 3xy2 + (m2 – 1) y1 = 0
Solution:
We have y = cos (m sin-1 x)
y1 = sin (m sin-1x). \(\frac{m}{\sqrt{1-x^{2}}}\)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.4 40