NCERT Solutions for Chapter 5 Continuity and Differentiability Class 12 Maths Exercise 5.7

Class 12 Maths NCERT Solutions for Chapter 5 Continuity and Differentiability Exercise 5.7

Continuity and Differentiability Exercise 5.7 Solutions

1. Find the second order derivatives of the function.
x² + 3x + 2

Solution

Let  y = x² + 3x + 2
Then, 

2. Find the second order derivative of the function x²⁰  .

Solution

Let, y = x²⁰ 
Then, 

3. Find  the second order derivatives of the function. 
x. cos x 

Solution

Let y = x . cos x 
Then, 

4.  Find the second order derivatives of the function. 
log x 

Solution

Let y = log x 
Then,  

5. Find the second order derivatives of the function. 
x³ log x 

Solution

Let y = x³ log x 
Then,  

= (1 + 3 log x) . 2x + x² . 3/x 
= 2x + 6x log x + 3x 
= 5x + 6x log x 
= x(5 + 6 log x)

6. Find the second order derivatives of the function. 
e^x sin 5x 

Solution

Let y = e^x sin 5x 
Then, 

= e^x (sin 5x + 5 cos 5x) + e^x (5cos 5x - 25 sin 5x)
= e^x (10 cos 5x - 24 sin 5x) = 2e^x (5 cos 5x - 12 sin 5x)

7. Find the second order derivatives of the function. 
e^6x cos 3x 

Solution

Let y = e^6x cos 3x 
Then, 

8. Find the second order derivatives of the function. 
tan^-1 x

Solution

Let y = tan^-1 x
Then, 

9. Find the second order derivatives of the function.
log (log x)

Solution

Let y = log (log x) 
Then, 

10. Find the second order derivatives of the function.
sin (log x)

Solution

Let y = sin(log x) 
The, 

11. If y = 5 cos x - 3 sin x, prove that (d² y/dx² ) + y = 0    . 

Solution

It is given that, y = 5 cos x - 3 sin x

Then

∴ (d² y/dx² ) + y = 0  
Hence, proved 

12. If y = cos^-1 x, Find d² y/dx²  in terms of y alone.

Solution

It is given that, y = cos^-1 x,
Then 

13. If y = 3 cos (log x) + 4 sin (log x), show that x² y₂ + xy₁ + y = 0

Solution

It is given that , y = 3 cos (log x) + 4 sin (log x),
Then, 

14. If y = Ae^mx + Be^nx , show that d² y/dx² - (m + n) dy/dx + mny = 0 

Solution

It is given that, y = Ae^mx + Be^nx 
Then,  

Hence, proved. 

15. If y = 500e^7x + 600e^-7x  , show that d² y/dx² = 49y

Solution

It is given that , y = = 500e^7x + 600e^-7x  
Then,  

16. If e^y (x + 1) = 1, show that d² y/dx²  = (dy/dx)² 

Solution

The given relationship is e^y (x + 1) = 1 
e^y (x + 1) = 1 
⇒ e^y = 1/(x + 1) 
Taking logarithm on both the sides, we obtain 
y = log . 1/(x + 1) 
Differentiating this relationship with respect to x, we obtain 

Hence, proved.

17. If y = (tan^–1 x)²  , show that (x² + 1)² y₂ + 2x (x² + 1) y₁ = 2

Solution

The given relationship is y = (tan^–1 x)²  
Then, 

Hence, proved.