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

In the previous exercises, you have learned about finding the first-order differentiation of different types of functions. In NCERT solutions for Class 12 Maths chapter 5 exercise 5.7, you will learn about finding the derivatives of second order. If a function y=f(x) is given, you can find its first-order derivative by differentiation of y y w.r.t x i.e. f'(x) = dy/dx and if you again differentiate f'(x) w.r.t the x, you will get the second-order f''(x).

If you have command on the first-order differentiation then exercise 5.7 Class 12 Maths will be very easy for you as you just need to differentiate the given function two times. The first and second-order derivatives are useful in finding minimum and maximum values of the functions, finding the domain and range of the function, other subjects also. If you looking for the NCERT solutions in one place, click on the above link. You will get NCERT solutions from Classes 6 to 12 for Science and Maths.

Also, see

  • Continuity and Differentiability Exercise 5.1
  • Continuity and Differentiability Exercise 5.2
  • Continuity and Differentiability Exercise 5.3
  • Continuity and Differentiability Exercise 5.4
  • Continuity and Differentiability Exercise 5.5
  • Continuity and Differentiability Exercise 5.6
  • Continuity and Differentiability Exercise 5.8
  • Continuity and Differentiability Miscellaneous Exercise

Continuity and Differentiability Exercise: 5.7

Question:1 Find the second order derivatives of the functions given in Exercises 1 to 10.

x^2 + 3x+ 2

Answer:

Given function is
y=x^2 + 3x+ 2
Now, differentiation w.r.t. x
\frac{dy}{dx}= 2x+3
Now, second order derivative
\frac{d^2y}{dx^2}= 2
Therefore, the second order derivative is \frac{d^2y}{dx^2}= 2

Question:2 Find the second order derivatives of the functions given in Exercises 1 to 10.

x ^{20}

Answer:

Given function is
y=x ^{20}
Now, differentiation w.r.t. x
\frac{dy}{dx}= 20x^{19}
Now, the second-order derivative is
\frac{d^2y}{dx^2}= 20.19x^{18}= 380x^{18}
Therefore, second-order derivative is \frac{d^2y}{dx^2}= 380x^{18}

Question:3 Find the second order derivatives of the functions given in Exercises 1 to 10.

x \cos x

Answer:

Given function is
y = x \cos x
Now, differentiation w.r.t. x
\frac{dy}{dx}= \cos x + x(-\sin x ) = \cos x-x\sin x
Now, the second-order derivative is
\frac{d^2y}{dx^2}= -\sin x-(\sin x+x\cos x) = -2\sin x - x\sin x
Therefore, the second-order derivative is \frac{d^2y}{dx^2}= -2\sin x - x\sin x

Question:4 Find the second order derivatives of the functions given in Exercises 1 to 10.

\log x

Answer:

Given function is
y=\log x
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{1}{x}
Now, second order derivative is
\frac{d^2y}{dx^2}= \frac{-1}{x^2}
Therefore, second order derivative is \frac{d^2y}{dx^2}= \frac{-1}{x^2}

Question:5 Find the second order derivatives of the functions given in Exercises 1 to 10.

x ^3 \log x

Answer:

Given function is
y=x^3\log x
Now, differentiation w.r.t. x
\frac{dy}{dx}=3x^2.\log x+x^3.\frac{1}{x}= 3x^2.\log x+ x^2
Now, the second-order derivative is
\frac{d^2y}{dx^2}= 6x.\log x+3x^2.\frac{1}{x}+2x=6x.\log x+3x+2x = x(6.\log x+5)
Therefore, the second-order derivative is \frac{d^2y}{dx^2} = x(6.\log x+5)

Question:6 Find the second order derivatives of the functions given in Exercises 1 to 10.

e ^x \sin5 x

Answer:

Given function is
y= e^x\sin 5x
Now, differentiation w.r.t. x
\frac{dy}{dx}=e^x.\sin 5x +e^x.5\cos 5x = e^x(\sin5x+5\cos5x)
Now, second order derivative is
\frac{d^2y}{dx^2}= e^x(\sin5x+5\cos5x)+e^x(5\cos5x+5.(-5\sin5x))
= e^x(\sin5x+5\cos5x)+e^x(5\cos5x-25\sin5x)=e^x(10\cos5x-24\sin5x)
=2e^x(5\cos5x-12\sin5x)
Therefore, second order derivative is \frac{dy}{dx}=2e^x(5\cos5x-12\sin5x)

Question:7 Find the second order derivatives of the functions given in Exercises 1 to 10.

e ^{6x}\cos 3x

Answer:

Given function is
y= e^{6x}\cos 3x
Now, differentiation w.r.t. x
\frac{dy}{dx}=6e^{6x}.\cos 3x +e^{6x}.(-3\sin 3x)= e^{6x}(6\cos 3x-3\sin 3x)
Now, second order derivative is
\frac{d^2y}{dx^2}= 6e^{6x}(6\cos3x-3\sin3x)+e^{6x}(6.(-3\sin3x)-3.3\cos3x)
= 6e^{6x}(6\cos3x-3\sin3x)-e^{6x}(18\sin3x+9\cos3x)
e^{6x}(27\cos3x-36\sin3x) = 9e^{6x}(3\cos3x-4\sin3x)
Therefore, second order derivative is \frac{dy}{dx} = 9e^{6x}(3\cos3x-4\sin3x)

Question:8 Find the second order derivatives of the functions given in Exercises 1 to 10.

\tan ^{-1} x

Answer:

Given function is
y = \tan^{-1}x
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{d(\tan^{-1}x)}{dx}=\frac{1}{1+x^2}
Now, second order derivative is
\frac{d^2y}{dx^2}= \frac{-1}{(1+x^2)^2}.2x = \frac{-2x}{(1+x^2)^2}
Therefore, second order derivative is \frac{d^2y}{dx^2} = \frac{-2x}{(1+x^2)^2}

Question:9 Find the second order derivatives of the functions given in Exercises 1 to 10.

\log (\log x )

Answer:

Given function is
y = \log(\log x)
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{d(\log(\log x))}{dx}=\frac{1}{\log x}.\frac{1}{x}= \frac{1}{x\log x}
Now, second order derivative is
\frac{d^2y}{dx^2}= \frac{-1}{(x\log x)^2}.(1.\log x+x.\frac{1}{x}) = \frac{-(\log x+1)}{(x\log x)^2}
Therefore, second order derivative is \frac{d^2y}{dx^2} = \frac{-(\log x+1)}{(x\log x)^2}

Question:10 Find the second order derivatives of the functions given in Exercises 1 to 10.

\sin (\log x )

Answer:

Given function is
y = \sin(\log x)
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{d(\sin(\log x))}{dx}=\cos (\log x).\frac{1}{x}= \frac{\cos (\log x)}{x}
Now, second order derivative is
Using Quotient rule
\frac{d^2y}{dx^2}=\frac{-\sin(\log x)\frac{1}{x}.x-\cos(\log x).1}{x^2} = \frac{-(\sin (\log x)+\cos(\log x))}{x^2}
Therefore, second order derivative is \frac{d^2y}{dx^2} = \frac{-(\sin (\log x)+\cos(\log x))}{x^2}

Question:11 If y = 5 \cos x - 3 \sin x prove that \frac{d^2y}{dx^2}+y = 0

Answer:

Given function is
y = 5 \cos x - 3 \sin x
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{d(5\cos x-3\sin x)}{dx}=-5\sin x-3\cos x
Now, the second-order derivative is
\frac{d^2y}{dx^2}=\frac{d^2(-5\sin x-3\cos x)}{dx^2}=-5\cos x+3\sin x
Now,
\frac{d^2y}{dx^2}+y=-5\cos x+3\sin x+5\cos x-3\sin x = 0
Hence proved

Question:12 If y = \cos ^{-1} x Find \frac{d ^2 y }{dx^2 } in terms of y alone.

Answer:

Given function is
y = \cos ^{-1} x
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{d( \cos ^{-1} x)}{dx}=\frac{-1}{\sqrt{1-x^2}}
Now, second order derivative is
\frac{d^2y}{dx^2}=\frac{d^2(\frac{-1}{\sqrt{1-x^2}})}{dx^2}=\frac{-(-1)}{(\sqrt{1-x^2})^2}.(-2x) = \frac{-2x}{1-x^2} -(i)
Now, we want \frac{d^2y}{dx^2} in terms of y
y = \cos ^{-1} x
x = \cos y
Now, put the value of x in (i)
\frac{d^2y}{dx^2} = \frac{-2\cos y }{1-\cos^2 y } = \frac{-2\cos y}{\sin ^2 y}= -2\cot y cosec y
(\because 1-\cos^2x =\sin^2 x\ and \ \frac{\cos x}{\sin x} = \cot x \ and \ \frac{1}{\sin x}= cosec x)
Therefore, answer is \frac{d^2y}{dx^2} = -2\cot y cosec y

Question:13 If y = 3 \cos (\log x) + 4 \sin (\log x), show that x^2 y_2 + xy_1 + y = 0

Answer:

Given function is
y = 3 \cos (\log x) + 4 \sin (\log x)
Now, differentiation w.r.t. x
y_1=\frac{dy}{dx}=\frac{d( 3 \cos (\log x) + 4 \sin (\log x))}{dx}=-3\sin(\log x).\frac{1}{x}+4\cos (\log x).\frac{1}{x}
=\frac{4\cos (\log x)-3\sin(\log x)}{x} -(i)
Now, second order derivative is
By using the Quotient rule
y_2=\frac{d^2y}{dx^2}=\frac{d^2(\frac{4\cos (\log x)-3\sin(\log x)}{x})}{dx^2}= \frac{(-4\sin(\log x).\frac{1}{x}-3\cos(\log x).\frac{1}{x}).x-1.(4\cos (\log x)-3\sin(\log x))}{x^2}
=\frac{-\sin(\log x)+7\cos (\log x)}{x^2} -(ii)
Now, from equation (i) and (ii) we will get y_1 \ and \ y_2
Now, we need to show
x^2 y_2 + xy_1 + y = 0
Put the value of y_1 \ and \ y_2 from equation (i) and (ii)
x^2\left ( \frac{-\sin(\log x)+7\cos (\log x)}{x^2} \right )+x\left ( \frac{4\cos (\log x)-3\sin(\log x)}{x} \right )+ 3 \cos (\log x) +4\sin(\log x)
-\sin(\log x)-7\cos(\log x)+4\cos(\log x)-3\sin(\log x)+3\ cos (\log x) +4\sin(\log x)
=0
Hence proved

Question:14 If y = A e ^{mx} + Be ^{nx} , show that \frac{d ^2 y}{dx^2} - (m+n) \frac{dy}{dx} + mny = 0

Answer:

Given function is
y = A e ^{mx} + Be ^{nx}
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{d(A e ^{mx} + Be ^{nx})}{dx}=mAe^{mx}+nBe^{nx} -(i)
Now, second order derivative is
\frac{d^2y}{dx^2}=\frac{d^2(mAe^{mx}+nBe^{nx})}{dx^2}= m^2Ae^{mx}+n^2Be^{nx} -(ii)
Now, we need to show
\frac{d ^2 y}{dx^2} - (m+n) \frac{dy}{dx} + mny = 0
Put the value of \frac{d^2y}{dx^2} \ and \ \frac{dy}{dx} from equation (i) and (ii)
m^2Ae^{mx}+n^2Be^{nx}-(m+n)(mAe^{mx}+nBx^{nx}) +mn(Ae^{mx}+Be^{nx})
m^2Ae^{mx}+n^2Be^{nx}-m^2Ae^{mx}-mnBx^{nx}-mnAe^{mx} -n^2Be^{nx}+mnAe^{mx}+mnBe^{nx}
=0
Hence proved

Question:15 If y = 500 e ^{7x} + 600 e ^{- 7x } , show that \frac{d^2 y}{dx ^2} = 49 y
Answer:

Given function is
y = 500 e ^{7x} + 600 e ^{- 7x }
Now, differentiation w.r.t. x
\frac{dy}{dx}=\frac{d(500 e ^{7x} + 600 e ^{- 7x })}{dx}=7.500e^{7x}-7.600e^{-7x} =3500e^{7x}-4200e^{-7x} -(i)
Now, second order derivative is
\frac{d^2y}{dx^2}=\frac{d^2(3500e^{7x}-4200e^{-7x})}{dx^2}
= 7.3500e^{7x}-(-7).4200e^{-7x}= 24500e^{7x}+29400e^{-7x} -(ii)
Now, we need to show
\frac{d^2 y}{dx ^2} = 49 y
Put the value of \frac{d^2y}{dx^2} from equation (ii)
24500e^{7x}+29400e^{-7x}=49(500e^{7x}+600e^{-7x})
= 24500e^{7x}+29400e^{-7x}
Hence, L.H.S. = R.H.S.
Hence proved

Question:16 If e ^y (x+1) = 1 show that \frac{d^2 y }{dx^2 } = (\frac{dy}{dx})^2

Answer:

Given function is
e ^y (x+1) = 1
We can rewrite it as
e^y = \frac{1}{x+1}
Now, differentiation w.r.t. x
\frac{d(e^y)}{dx}=\frac{d(\frac{1}{x+1})}{dx}\\ e^y.\frac{dy}{dx}= \frac{-1}{(x+1)^2}\\ \frac{1}{x+1}.\frac{dy}{dx}= \frac{-1}{(x+1)^2} \ \ \ \ \ \ \ \ \ (\because e^y = \frac{1}{x+1})\\ \frac{dy}{dx}= \frac{-1}{x+1} -(i)
Now, second order derivative is
\frac{d^2y}{dx^2}=\frac{d^2(\frac{-1}{x+1})}{dx^2}=\frac{-(-1)}{(x+1)^2} = \frac{1}{(x+1)^2} -(ii)
Now, we need to show
\frac{d^2 y }{dx^2 } = (\frac{dy}{dx})^2
Put value of \frac{d^2y}{dx^2} \ and \ \frac{dy}{dx} from equation (i) and (ii)
\frac{1}{(x+1)^2}=\left ( \frac{-1}{x+1} \right )^2
=\frac{1}{(x+1)^2}
Hence, L.H.S. = R.H.S.
Hence proved

Question:17 If y = (\tan^{-1} x)^2 show that (x^2 + 1)^2 y_2 + 2x (x^2 + 1) y_1 = 2

Answer:

Given function is
y = (\tan^{-1} x)^2
Now, differentiation w.r.t. x
y_1=\frac{dy}{dx}=\frac{d((\tan^{-1}x)^2)}{dx}= 2.\tan^{-1}x.\frac{1}{1+x^2}= \frac{2\tan^{-1}x}{1+x^2} -(i)
Now, the second-order derivative is
By using the quotient rule
y_2=\frac{d^2y}{dx^2}=\frac{d^2(\frac{2\tan^{-1}x}{1+x^2})}{dx^2}=\frac{2.\frac{1}{1+x^2}.(1+x^2)-2\tan^{-1}x(2x)}{(1+x^2)^2}=\frac{2-4x\tan^{-1}x}{(1+x^2)^2} -(ii)
Now, we need to show
(x^2 + 1)^2 y_2 + 2x (x^2 + 1) y_1 = 2
Put the value from equation (i) and (ii)
(x^2+1)^2.\frac{2-4x\tan^{-1}x}{(1+x^2)^2}+2x(x^2+1).\frac{2\tan^{-1}x}{x^2+1}\\ \Rightarrow 2-4x\tan^{-1}x+4x\tan^{-1}x = 2
Hence, L.H.S. = R.H.S.
Hence proved

More About NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.7

In Class 12th Maths chapter 5 exercise 5.7, you will get 17 questions related to finding the second-order derivatives. There are four examples given in the NCERT book prior to the ex 5.7 which you can solve to get more familiar with second derivatives before solving the exercise question. After solving examples, you can try to solve Class 12th Maths chapter 5 exercise 5.7 questions. You may not be to solve these exercise 5.7 Class 12 Maths problems by yourself at first. You can go through Class 12 Maths chapter 5 exercise 5.7 solutions to get conceptual clarity.

Also Read| Continuity and Differentiability Class 12th Chapter 5 Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.7

  • Class 12 Maths chapter 5 exercise 5.7 solutions are designed in a descriptive manner that you can understand easily.
  • Don't go through NCERT solutions for Class 12 Maths chapter 5 exercise 5.7 without trying to solve NCERT problems by yourself.
  • You can use exercise 5.7 Class 12 Maths solutions for reference.

Also see-

  • NCERT Solutions for Class 12 Maths Chapter 5

  • NCERT Exemplar Solutions Class 12 Maths Chapter 5

NCERT Solutions of Class 12 Subject Wise

  • NCERT Solutions for Class 12 Maths

  • NCERT Solutions for Class 12 Physics

  • NCERT Solutions for Class 12 Chemistry

  • NCERT Solutions for Class 12 Biology

Subject Wise NCERT Exampler Solutions

  • NCERT Exemplar Solutions for Class 12th Maths

  • NCERT Exemplar Solutions for Class 12th Physics

  • NCERT Exemplar Solutions for Class 12th Chemistry

  • NCERT Exemplar Solutions for Class 12th Biology

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