Continuity & Differentiability(Ex. 5.2)

 Exercise 5.2

Differentiate the functions with respect to x in Exercises 1 to 8.

1. sin (x2+5)

Solun:- Let y = sin (x2+5)

Let t = x2+5

⇒ y = sin t

Differentiate w.r.t. x:-

⇒ t = x2+5

Differentiate w.r.t. x:-

2. cos (sin x)

Solun:- Let y = cos (sin x)

Let t = sin x

⇒ y = cos t

Differentiate w.r.t. x:-

⇒ t = sin x

Differentiate w.r.t. x:-

3. sin (ax+b)

Solun:- Let y = sin (ax+b)

Let t = ax+b

⇒ y = sin t

Differentiate w.r.t. x:-

⇒ t = ax+b

Differentiate w.r.t. x:-

4. sec (tan√x)

Solun:- Let y = sec (tan√x)

Let t = tan√x

⇒ y = sec t

Differentiate w.r.t. x:-

⇒ t = tan√x

Differentiate w.r.t. x:-

Let u = sin (ax+b)  and  v = cos (cx+d)

Differentiate w.r.t. x:-

6. cos x3 . sin2 (x5)

Solun:- Let y = cos x3 . sin2 (x5)

Let u = cos x3  and  v = sin2 (x5)

Differentiate w.r.t. x:-

7. 2√cot(x2)

Solun:- Let y = 2√cot(x2)

Differentiate w.r.t. x:-

Applying Chain Rule:-

8. cos(√x)

Solun:- Let y = cos(√x)

Differentiate w.r.t. x:-

Applying Chain Rule:-

9. Prove that the function f given by f(x) = |x - 1|, x∈R is not differentiable at x = 1.

Solun:- Let f(x) = |x - 1|

We know for differentiability:-

R.H.D = L.H.D

R.H.D:-  

At x>1

Put h = 0

= 1

L.H.D:-  

At x<1

Put h = 0

= - 1

R.H.D ≠ L.H.D

Hence given function is not differentiable at x = 1.

10. Prove that the greater integer function defined by f(x) = [x], 0<x<3 is not differentiable at x = 1 and at x=2.

Solun:- Let f(x) = [x]

We know for differentiability:-

R.H.D = L.H.D

At x = 1

R.H.D:-  

Put h = 0

= 0

L.H.D:-

Put h = 0

L.H.D is not defined.

Hence given function is not differentiable at x = 1.

At x = 2

R.H.D:-  

Put h = 0

= 0

L.H.D:-

Put h = 0

L.H.D is not defined.

Hence given function is not differentiable at x = 2.

Download PDF of Exercise 5.2

SEE ALSO:-

Notes of Continuity & Differentiability

Exercise 5.1

If you have any queries, you can ask me in the comment section

And you can follow/subscribe me for the latest updates on your e-mails

For subscribing me follow these instructions:-

1. Fill your E-mail address

2. Submit Recaptcha

3. Go to your email and then click on the verify link

Then you get all update on your email

Thanks for Reading ......