Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 10 Differentiability and Methods of Differentiation Ex 10.2 Text Book Back Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.2

Find the derivatives of the following functions with respect to corresponding independent variables:

Question 1.
f(x) = x – 3 sin x
f(x) = x – 3 sinx
= f'(x) = 1 – 3 (cos x)
= 1 – 3 cos x

Question 2.
y = sin x + cos x
y = sin x + cos x
$$\frac{d y}{d x}$$ = cos x – sin x

Question 3.
f(x) = x sin x
f(x) = uv
⇒ f'(x) = uv’ + vu’ = u$$\frac{d u}{d x}$$ + v$$\frac{d v}{d x}$$
Now u = x ⇒ u’ = 1
v = sin x ⇒ v’ cos x
f'(x) = x (cos x) + sin x(1)
= x cos x + sin x

Question 4.
y = cos x – 2 tan x
y = cos x – 2 tan x
$$\frac{d y}{d x}$$ = – sin x – 2 sec2 x

Question 5.
g(t) = t3 cos t
g(t) = t3 cost (i.e.) u = t3 and v = cos t
let u’ = $$\frac{d u}{d x}$$ and v’= $$\frac{d v}{d x}$$ = (-sint)
g'(t) = uv’ + vu’
g'(t) = t3 (-sin t) + cos t (3t2)
= -t3 sin t + 3t2 cos t

Question 6.
g(t) = 4 sec t + tan t
g(t) = 4 sec t + tan t
g'(t) = 4 sec t tan t + sec2t

Question 7.
y = ex sin x
y = ex sin x
⇒ y = uv’ + vu’
Now u = ex ⇒ u’ = $$\frac{d u}{d x}$$ ex
v = sin x ⇒ v’ = $$\frac{d v}{d x}$$ cos x
i.e. y’ = ex (cos x) + sin x (ex)
= ex [sin x + cos x]

Question 8.
y = $$\frac{\tan x}{x}$$

Question 9.
y = $$\frac{\sin x}{1+\cos x}$$

Question 10.
y = $$\frac{x}{\sin x+\cos x}$$

Question 11.
y = $$\frac{\tan x-1}{\sec x}$$

Question 12.
y = $$\frac{\sin x}{x^{2}}$$

Question 13.
y = tan θ (sin θ + cos θ)
y = tan θ (sin θ + cos θ)
$$\frac{d y}{d x}$$ = tan θ (cos θ – sin θ) + (sin θ + cos 0) sec2 θ
= tan θ cos θ – tan θ sin θ + sin θ sec2θ + cos θ sec2θ

= sin θ – sin2θ sec θ + tan θ sec θ + sec θ
= sin θ + (1 – sin2θ) sec θ + sec θ tan θ
= sin θ + cos2θ × $$\frac{1}{\cos \theta}$$ + sec θ tan θ
= sin θ + cos θ + sec θ tan θ

Question 14.
y = cosec x . cot x
y = cosec x . cot x
$$\frac{d y}{d x}$$ = cosec x × – cosec2 x + cot x × – cosec x cot x
= – cosec3 x – cosec x cot2 x
= – cosec x (cosec2 x + cot2 x)

Question 15.
y = x sin x cos x

Question 16.
y = e-x . log x
y = e-x . log x
$$\frac{d y}{d x}$$ = e-x × $$\frac{1}{x}$$ + (log x) e-x (-1)

Question 17.
y = (x2 + 5) log (1 + x) e-3x

Question 18.
y = sin x0