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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3

Differentiate the following
Question 1.
y = (x2 + 4x + 6)5
Solution:
Let = u = x2 + 4x + 6
⇒ \(\frac{d u}{d x}\) = 2x + 4
Now y = u5 ⇒ \(\frac{d y}{d x}\) = 5u4
∴ \(\frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x}\) = 5u4 (2x + 4)
= 5(x2 + 4x + 6)4 (2x + 4)
= 5 (2x + 4) (x2 + 4x + 6)4

Question 2.
y = tan 3x
Solution:
y = tan 3x
put u = 3x
\(\frac{d u}{d x}\) = 3
Now y = tan u
⇒ \(\frac{d u}{d x}\) = sec2 u
So \(\frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x}\) = (sec2 u) (3)
= 3 sec2 3x

Question 3.
y = cos (tan x)
Solution:
Put u = tan x
\(\frac{d u}{d x}\) = sec2x
Now y = cos u ⇒ \(\frac{d u}{d x}\) = – sin u
Now \(\frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x}\)
= (-sin u) (sec2x)
= – sec2 (sin (tan x))

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3

Question 4.
y = \(\sqrt[3]{1+x^{3}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 1
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 2

Question 5.
y = \(e^{\sqrt{x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 3

Question 6.
y = sin (ex)
Solution:
y = sin (ex)
Let u = ex
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 4

Question 7.
F(x) = (x3 + 4x)7
Solution:
F(x) = (x3 + 4x)7
Put u = x3 + 4x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 5

Question 8.
h(t) = \(\left(t-\frac{1}{t}\right)^{\frac{3}{2}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 6

Question 9.
f(t) = \(\sqrt[3]{1+\tan t}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 7

Question 10.
y = cos (a3 + x3)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 8

Question 11.
y = e-mx
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 9

Question 12.
y = 4 sec 5x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 10

Question 13.
y = (2x – 5)4 (8x2 – 5)-3
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 11
= \(\frac{8(2 x-5)^{3}}{\left(8 x^{2}-5\right)^{4}}\) (-4x2 + 30x – 5)

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3

Question 14.
y = (x2 + 1) \(\sqrt[3]{x^{2}+2}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 12
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 13

Question 15.
y = xe-x2
Solution:
y = xe-x2
y = uv where u = x and v = e-x2
Now u’ = 1 and v’ = e-x2 (-2x)
v’ = – 2xe-x2
Now y = uv ⇒ y’ = uv’ + vu’
(i.e.) \(\frac{d y}{d x}\) = x[-2xe-x2] + e-x2 (1)
= e-x2 (1 – 2x2)

Question 16.
s(t) = \(\sqrt[4]{\frac{t^{3}+1}{t^{3}-1}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 14
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 15

Question 17.
f(x) = \(\frac{x}{\sqrt{7-3 x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 16
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 17

Question 18.
y = tan (cos x)
Solution:
y = tan (cos x)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 18

Question 19.
y = \(\frac{\sin ^{2} x}{\cos x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 19

Question 20.
y = \(5^{-\frac{1}{x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 20

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3

Question 21.
y = \(\sqrt{1+2 \tan x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 21

Question 22.
y = sin3x + cos3x
Solution:
y = sin3x + cos3x
Here u = sin3 x = (sin x)3
⇒ \(\frac{d u}{d x}\) = 3 (sin x)2 (cos x)
= 3sin2x cos x
v = cos3x = (cos x)3
⇒ \(\frac{d v}{d x}\) = 3 (cos x)2 (-sin x) = -3 sin x cos2x
Now y = u + v ⇒ \(\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}\)
= 3 sin2x cos x – 3sin x cos2x
= 3 sin x cos x (sin x – cos x)

Question 23.
y = sin2 (cos kx)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 22

Question 24.
y = (1 + cos2x)6
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 23

Question 25.
y =\(\frac{e^{3 x}}{1+e^{x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 24
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 24

Question 26.
y = \(\sqrt{x+\sqrt{x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 26

Question 27.
y = ex cos x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 27

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3

Question 28.
y = \(\sqrt{x+\sqrt{x+\sqrt{x}}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 28
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 29

Question 29.
y = \(\sin (\tan (\sqrt{\sin x}))\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 30

Question 30.
y = sin-1\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 31
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.3 32

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