Exercise 5.7

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

1. x2+3x+2

Solun:- Let y = x2+3x+2
Differentiate w.r.t. x:-
${"type":"align","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(x^{2}+3x+2\\right)}{x}}\\\\\n{\\diff{y}{x}}&={2x+3}\t\n\\end{align*}","id":"1-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1598779932890,"cs":"cbvWworI64ENL1iJGKW/qw==","size":{"width":154,"height":74}}$
Again Differentiate w.r.t. x:-
${"font":{"family":"Arial","size":12,"color":"#000000"},"id":"2-0","type":"$","code":"$\\frac{d^{2}y}{dx^{2}}=2$","ts":1598780223901,"cs":"7XcPAr/HtaWHhykMNGYY1A==","size":{"width":68,"height":29}}$

2. x20

Solun:- Let y = x20
Differentiate w.r.t. x:-
${"id":"1-1-0-0","type":"align","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(x^{20}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={20x^{19}}\t\n\\end{align}","ts":1598780411165,"cs":"A+kkDW6wVF3XGFYSTHvlnw==","size":{"width":96,"height":74}}$
Again Differentiate w.r.t. x:-
${"id":"2-1-0-0","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align","code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={20\\times19x^{18}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={380x^{18}}\t\n\\end{align}","ts":1598780470203,"cs":"hyknk4FPNLKjGy7tY7QevQ==","size":{"width":124,"height":76}}$

3. x.cos x

Solun:- Let y = x.cos x
Differentiate w.r.t. x:-
${"type":"align","font":{"color":"#000000","family":"Arial","size":10},"id":"1-1-1","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(x.\\cos x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={x.\\diff{\\left(\\cos x\\right)}{x}+\\cos x\\diff{x}{x}}\\\\\n{\\diff{y}{x}}&={-x.\\sin x+\\cos x}\t\n\\end{align}","ts":1598780852720,"cs":"+MjFiIHZ5epB2rBNvzg5Iw==","size":{"width":200,"height":110}}$
Again Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(-x.\\sin x+\\cos x\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-\\left(x.\\diff{\\sin x}{x}+\\sin x\\diff{x}{x}\\right)+\\diff{\\left(\\cos x\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-x.\\cos x-\\sin x-\\sin x}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-x.\\cos x-2\\sin x}\t\n\\end{align}","id":"2-1-1","type":"align","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1598781177858,"cs":"XfOV1gafrEzN68lOm0xS8Q==","size":{"width":308,"height":160}}$
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(-x.\\sin x+\\cos x\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-\\left(x.\\diff{\\sin x}{x}+\\sin x\\diff{x}{x}\\right)+\\diff{\\left(\\cos x\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-x.\\cos x-\\sin x-\\sin x}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-x.\\cos x-2\\sin x}\t\n\\end{align}","id":"2-1-1","type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1598781177858,"cs":"XfOV1gafrEzN68lOm0xS8Q==","size":{"width":308,"height":160}}$

4. log x

Solun:- Let y = log x
Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(\\log_{}x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{1}{x}}\t\n\\end{align}","id":"1-1-0-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align","ts":1598781326756,"cs":"Lkt7a2mlVP8Fc+5+bDgbvQ==","size":{"width":106,"height":72}}$
Again Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(\\frac{1}{x}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-1}{x^{2}}}\t\n\\end{align}","type":"align","id":"2-1-0-1-0-0","font":{"family":"Arial","color":"#000000","size":10},"ts":1598781379296,"cs":"lKO0ZAJTz4h50+KFqv1Gxw==","size":{"width":94,"height":77}}$

5. x3log x

Solun:- Let y = x3log x
Differentiate w.r.t. x:-
${"font":{"family":"Arial","color":"#000000","size":10},"id":"1-1-0-2","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(x^{3}.\\log_{}x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={x^{3}\\diff{\\left(\\log_{}x\\right)}{x}+\\log_{}x.\\diff{\\left(x^{3}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={x^{3}\\times\\frac{1}{x}+3x^{2}.\\log_{}x}\\\\\n{\\diff{y}{x}}&={x^{2}+3x^{2}.\\log_{}x}\t\n\\end{align}","type":"align","ts":1598781539541,"cs":"8pE1S9gZONnz+SAZd5so/A==","size":{"width":226,"height":153}}$
Again Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(x^{2}+3x^{2}.\\log_{}x\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={2x+\\left(3x^{2}\\diff{\\left(\\log_{}x\\right)}{x}+\\log_{}x.\\diff{\\left(3x^{2}\\right)}{x}\\right)}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={2x+\\left(3x^{2}\\times\\frac{1}{x}+6x.\\log_{}x\\right)}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={2x+3x+6x.\\log_{}x}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={5x+6x.\\log_{}x}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={x\\left(5+6\\log_{}x\\right)}\t\n\\end{align}","type":"align","font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-1-0-1-1","ts":1598782031103,"cs":"YZhhCliUx+6B9ONVje3UWg==","size":{"width":309,"height":254}}$

6. exsin 5x

Solun:- Let y = exsin 5x
Differentiate w.r.t. x:-
${"type":"align","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(e^{x}.\\sin5x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={e^{x}\\diff{\\left(\\sin5x\\right)}{x}+\\sin5x\\diff{\\left(e^{x}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={5e^{x}\\cos5x+\\sin5x.e^{x}}\\\\\n{\\diff{y}{x}}&={e^{x}\\left(5\\cos5x+\\sin5x\\right)}\t\n\\end{align}","id":"1-1-0-1-1-0-0","ts":1598782321191,"cs":"rfd1E2I0kOB8us13IdoSRw==","size":{"width":228,"height":148}}$
Again Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left[e^{x}\\left(5\\cos5x+\\sin5x\\right)\\right]}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={e^{x}\\diff{\\left(5\\cos5x+\\sin5x\\right)}{x}+\\left(5\\cos5x+\\sin5x\\right)\\diff{\\left(e^{x}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={e^{x}\\left(-25\\sin5x+5\\cos5x\\right)+e^{x}\\left(5\\cos5x+\\sin5x\\right)}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={e^{x}\\left(-25\\sin5x+5\\cos5x+5\\cos5x+\\sin5x\\right)}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={e^{x}\\left(-24\\sin5x+10\\cos5x\\right)}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={2e^{x}\\left(-12\\sin5x+5\\cos5x\\right)}\t\n\\end{align}","font":{"family":"Arial","color":"#000000","size":10},"id":"2-1-0-1-0-1-0-0","type":"align","ts":1598782629387,"cs":"9R1R+XMfGRqDWJ9gDpYU4Q==","size":{"width":388,"height":236}}$

7. e6xcos 3x

Solun:- Let y = e6xcos 3x
Differentiate w.r.t. x:-
${"font":{"color":"#000000","family":"Arial","size":10},"type":"align","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(e^{6x}.\\cos 3x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={e^{6x}\\diff{\\left(\\cos3x\\right)}{x}+\\cos3x\\diff{\\left(e^{6x}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={-3e^{6x}.\\sin3x+6e^{6x}.\\cos3x}\\\\\n{\\diff{y}{x}}&={3e^{6x}\\left(-\\sin3x+2\\cos3x\\right)}\t\n\\end{align}","id":"1-1-0-1-1-0-1","ts":1598783723825,"cs":"Qyr7eSLufd9gOAAWzAj/XA==","size":{"width":245,"height":153}}$
Again Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left[3e^{6x}\\left(-\\sin 3x+2\\cos3x\\right)\\right]}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3\\left[e^{6x}\\diff{\\left(-\\sin 3x+2\\cos3x\\right)}{x}+\\left(-\\sin 3x+2\\cos3x\\right)\\diff{\\left(e^{6x}\\right)}{x}\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3\\left[e^{6x}\\left(-3\\cos3x-6\\sin3x\\right)+6e^{6x}\\left(-\\sin 3x+2\\cos3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3e^{6x}\\left[\\left(-3\\cos3x-6\\sin3x\\right)+6\\left(-\\sin 3x+2\\cos3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3e^{6x}\\left[\\left(-3\\cos3x-6\\sin3x\\right)+\\left(-6\\sin 3x+12\\cos3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3e^{6x}\\left[\\left(9\\cos3x-12\\sin3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={9e^{6x}\\left[\\left(3\\cos3x-4\\sin3x\\right)\\right]}\t\n\\end{align}","id":"2-1-0-1-0-1-0-1","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align","ts":1598784930824,"cs":"5Ywjp2lYlIBSGaOdxADIUw==","size":{"width":457,"height":290}}$
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left[3e^{6x}\\left(-\\sin 3x+2\\cos3x\\right)\\right]}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3\\left[e^{6x}\\diff{\\left(-\\sin 3x+2\\cos3x\\right)}{x}+\\left(-\\sin 3x+2\\cos3x\\right)\\diff{\\left(e^{6x}\\right)}{x}\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3\\left[e^{6x}\\left(-3\\cos3x-6\\sin3x\\right)+6e^{6x}\\left(-\\sin 3x+2\\cos3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3e^{6x}\\left[\\left(-3\\cos3x-6\\sin3x\\right)+6\\left(-\\sin 3x+2\\cos3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3e^{6x}\\left[\\left(-3\\cos3x-6\\sin3x\\right)+\\left(-6\\sin 3x+12\\cos3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3e^{6x}\\left[\\left(9\\cos3x-12\\sin3x\\right)\\right]}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={9e^{6x}\\left[\\left(3\\cos3x-4\\sin3x\\right)\\right]}\t\n\\end{align}","id":"2-1-0-1-0-1-0-1","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","ts":1598784930824,"cs":"5Ywjp2lYlIBSGaOdxADIUw==","size":{"width":457,"height":290}}$

8. tan-1x

Solun:- Let y = tan-1x
Differentiate w.r.t. x:-
${"font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(\\tan^{-1}x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{1}{1+x^{2}}}\t\n\\end{align}","type":"align","id":"1-1-0-1-1-1-0","ts":1598783174873,"cs":"GI+ej1t9soaAiUY4WadCvQ==","size":{"width":125,"height":76}}$
Again Differentiate w.r.t. x:-
${"type":"align","id":"2-1-0-1-0-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(\\frac{1}{1+x^{2}}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{\\left(1+x^{2}\\right)\\diff{\\left(1\\right)}{x}-1\\times\\diff{\\left(1+x^{2}\\right)}{x}}{\\left(1+x^{2}\\right)^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{\\left(1+x^{2}\\right)\\times0-2x}{\\left(1+x^{2}\\right)^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-2x}{\\left(1+x^{2}\\right)^{2}}}\t\n\\end{align}","ts":1598783400195,"cs":"2PHyBY0cwqxrRVyJg4dO4Q==","size":{"width":238,"height":198}}$

9. log(log x)

Solun:- Let y = log(log x)
Differentiate w.r.t. x:-
${"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(\\log_{}\\left(\\log_{}x\\right)\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{1}{\\log_{}x}\\times\\frac{1}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{1}{x.\\log_{}x}}\t\n\\end{align}","id":"1-1-0-1-1-1-1","type":"align","ts":1598785484706,"cs":"cyzUjZqQadw6samEhKPL0Q==","size":{"width":140,"height":116}}$
Again Differentiate w.r.t. x:-
${"id":"2-1-0-1-0-1-1-1-0","code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(\\frac{1}{x.\\log_{}x}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{\\left(x.\\log_{}x\\right)\\diff{\\left(1\\right)}{x}-1\\times\\diff{\\left(x.\\log_{}x\\right)}{x}}{\\left(x.\\log_{}x\\right)^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{\\left(x.\\log_{}x\\right)\\times0-\\left(x.\\diff{\\left(\\log_{}x\\right)}{x}+\\log_{}x.\\diff{x}{x}\\right)}{\\left(x.\\log_{}x\\right)^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-\\left(x\\times\\frac{1}{x}+\\log_{}x\\right)}{\\left(x.\\log_{}x\\right)^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-\\left(1+\\log_{}x\\right)}{\\left(x.\\log_{}x\\right)^{2}}}\t\n\\end{align}","type":"align","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1598785729174,"cs":"pnJx/9q640USTMUyxw2Qgw==","size":{"width":324,"height":254}}$

10. sin(log x)

Solun:- Let y = sin(log x)
Differentiate w.r.t. x:-
${"type":"align","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(\\sin\\left(\\log_{}x\\right)\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\cos\\left(\\log_{}x\\right)\\times\\frac{1}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{\\cos\\left(\\log_{}x\\right)}{x}}\t\n\\end{align}","id":"1-1-0-1-1-1-2-0","ts":1598785921239,"cs":"le8s7Pr6Sq3ps6jOvkNvTA==","size":{"width":148,"height":110}}$
Again Differentiate w.r.t. x:-
${"id":"2-1-0-1-0-1-1-1-1-0","type":"align","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(\\frac{\\cos\\left(\\log_{}x\\right)}{x}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{\\left(x\\right)\\diff{\\left(\\cos\\left(\\log_{}x\\right)\\right)}{x}-\\cos\\left(\\log_{}x\\right)\\times\\diff{\\left(x\\right)}{x}}{x^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-x\\times \\sin\\left(\\log_{}x\\right)\\times\\frac{1}{x}-\\cos\\left(\\log_{}x\\right)}{x^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-\\sin\\left(\\log_{}x\\right)-\\cos\\left(\\log_{}x\\right)}{x^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-\\left[\\frac{\\sin\\left(\\log_{}x\\right)+\\cos\\left(\\log_{}x\\right)}{x^{2}}\\right]}\t\n\\end{align}","ts":1598970372618,"cs":"Oarfoy923Id/MeQpla2wqQ==","size":{"width":281,"height":220}}$

11. If y = 5cos x - 3sin x, prove that

${"id":"3-0","code":"$\\frac{d^{2}y}{dx^{2}}+y=0$","type":"$","font":{"color":"#000000","family":"Arial","size":11},"ts":1598856571923,"cs":"A9vHbqWipqusf6diR5ISiA==","size":{"width":90,"height":25}}$
Solun:- Let y = 5cos x - 3sin x
Differentiate w.r.t. x:-
${"type":"align","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(5\\cos x-3\\sin x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={-5\\sin x-3\\cos x}\\\\\n{\\diff{y}{x}}&={-\\left(5\\sin x+3\\cos x\\right)}\t\n\\end{align}","font":{"family":"Arial","color":"#000000","size":10},"id":"1-1-0-1-1-1-2-1-0","ts":1598856675970,"cs":"KWzocPLmIlqmrW7waSyv1g==","size":{"width":178,"height":108}}$
Again Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(-\\left(5\\sin x+3\\cos x\\right)\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-\\left(5\\cos x-3\\sin x\\right)}\t\n\\end{align}","font":{"family":"Arial","color":"#000000","size":10},"id":"2-1-0-1-0-1-1-1-1-1-0","type":"align","ts":1598856766612,"cs":"V66H0IYc2O89N3gqlNcZKg==","size":{"width":208,"height":76}}$
Given 5cos x - 3sin x = y
${"font":{"color":"#000000","family":"Arial","size":12},"code":"$\\frac{d^{2}y}{dx^{2}}=-y$","type":"$","id":"4-0","ts":1598856840556,"cs":"kuDOolX4AMnYiP7uO2qtSg==","size":{"width":82,"height":29}}$
${"type":"$","code":"$\\frac{d^{2}y}{dx^{2}}+y=0$","id":"5-0","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1598856865937,"cs":"gMSfbfLGAEZNJ2YZbxfyow==","size":{"width":102,"height":29}}$ (Hence Proved....)

12. If y = cos-1x, prove that ${"font":{"size":11,"color":"#000000","family":"Arial"},"type":"$","id":"3-1","code":"$\\frac{d^{2}y}{dx^{2}}$","ts":1598865171217,"cs":"vl03l6KKsyUBZwfNzrJP+g==","size":{"width":26,"height":25}}$ in terms of y alone.

Solun:- Let y = cos-1x
Differentiate w.r.t. x:-
${"font":{"family":"Arial","color":"#000000","size":10},"id":"1-1-0-1-1-1-2-1-1-0","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(\\cos ^{-1}x\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\frac{-1}{{\\sqrt[]{1-x^{2}}}}}\\\\\n\\end{align}","type":"align","ts":1598865294203,"cs":"91rNuhZ3kzbEGGYvK9y4Cg==","size":{"width":124,"height":80}}$
Again Differentiate w.r.t. x:-
${"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(\\frac{-1}{{\\sqrt[]{1-x^{2}}}}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{{\\sqrt[]{1-x^{2}}}\\diff{\\left(-1\\right)}{x}-\\left(-1\\right)\\diff{\\left({\\sqrt[]{1-x^{2}}}\\right)}{x}}{\\left(1-x^{2}\\right)}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{{\\sqrt[]{1-x^{2}}}\\times0+\\frac{1}{2{\\sqrt[]{1-x^{2}}}}\\times\\left(-2x\\right)}{\\left(1-x^{2}\\right)}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-x}{\\left(1-x^{2}\\right){\\sqrt[]{1-x^{2}}}}}\t\n\\end{align}","id":"2-1-0-1-0-1-1-1-1-1-1-0","type":"align","ts":1598866079550,"cs":"/PaCH4daMJbjRvtT7sxwBA==","size":{"width":269,"height":204}}$
Given y = cos-1x ⇒ x = cos y
${"font":{"color":"#000000","size":7.890625,"family":"Arial"},"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-\\cos y}{\\left(1-\\cos^{2}y\\right){\\sqrt[]{1-\\cos^{2}y}}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-\\cos y}{\\sin^{2}y\\times \\sin y}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={-\\cot y.\\cos ec^{2}y}\t\n\\end{align}","id":"4-1-0","type":"align*","ts":1598866493894,"cs":"DW14nTydiqNMr7tEHPFDhQ==","size":{"width":216,"height":106}}$

13. If y = 3cos(log x) + 4sin(log x), show that x2y2 + xy1 + y = 0

Solun:- Let y = 3cos(log x) + 4sin(log x)
Differentiate w.r.t. x:-
${"type":"align","id":"1-1-0-1-1-1-2-1-1-1-0","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(3\\cos \\left(\\log_{}x\\right)+4\\sin\\left(\\log_{}x\\right)\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\left[-3\\sin\\left(\\log_{}x\\right)\\times\\frac{1}{x}+4\\cos\\left(\\log_{}x\\right)\\times\\frac{1}{x}\\right]}\\\\\n{y_{1}}&={\\frac{\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]}{x}......\\left(1\\right)}\t\n\\end{align}","font":{"color":"#000000","family":"Arial","size":10},"ts":1598869036930,"cs":"GsvaTs+fhZHNoM4AferHUQ==","size":{"width":316,"height":116}}$
Again Differentiate w.r.t. x:-
${"id":"2-1-0-1-0-1-1-1-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":8},"type":"align","code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(\\frac{\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]}{x}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{x.\\diff{\\left(\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]\\right)}{x}-\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]\\diff{\\left(x\\right)}{x}}{x^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{x.\\left[-4\\sin\\left(\\log_{}x\\right)\\times\\frac{1}{x}-3\\cos\\left(\\log_{}x\\right)\\times\\frac{1}{x}\\right]-\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]}{x^{2}}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\frac{-x\\times\\frac{1}{x}.\\left[4\\sin\\left(\\log_{}x\\right)+3\\cos\\left(\\log_{}x\\right)\\right]-\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]}{x^{2}}}\\\\\n{y_{2}}&={\\frac{-\\left[4\\sin\\left(\\log_{}x\\right)+3\\cos\\left(\\log_{}x\\right)\\right]-\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]}{x^{2}}}\\\\\n{x^{2}y_{2}}&={-\\left[4\\sin\\left(\\log_{}x\\right)+3\\cos\\left(\\log_{}x\\right)\\right]-\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]}\t\n\\end{align}","ts":1598868791066,"cs":"53HRijUOKJ33MRuLtysHGQ==","size":{"width":460,"height":206}}$
Given y = 3cos(log x) + 4sin(log x)
From Eq. 1:-
${"code":"\\begin{align}\n{x^{2}y_{2}}&={-\\left[4\\sin\\left(\\log_{}x\\right)+3\\cos\\left(\\log_{}x\\right)\\right]-\\left[4\\cos\\left(\\log_{}x\\right)-3\\sin\\left(\\log_{}x\\right)\\right]}\\\\\n{x^{2}y_{2}}&={-y-xy_{1}}\t\n\\end{align}","font":{"family":"Arial","size":7.890625,"color":"#000000"},"type":"align*","id":"4-1-1-0","ts":1598971165163,"cs":"b1S/7toIYLiJoyA2kkBdgw==","size":{"width":378,"height":34}}$
x2y2 + xy1 + y = 0 (Hence Proved...)

14. If y = Aemx + Benx , show that

${"code":"$\\frac{d^{2}y}{dx^{2}}-\\left(m+n\\right)\\diff{y}{x}+mny=0$","id":"7-0","font":{"size":11,"color":"#000000","family":"Arial"},"type":"$","ts":1598869013204,"cs":"X7NdTFYHSEjRRiyYttd23g==","size":{"width":225,"height":25}}$
Solun:- Let y = Aemx + Benx
Differentiate w.r.t. x:-
${"font":{"color":"#000000","family":"Arial","size":10},"type":"align","id":"1-1-0-1-1-1-2-1-1-1-1-0","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(Ae^{mx}+ Be^{nx}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\left(Ame^{mx}+Bne^{nx}\\right).....\\left(1\\right)}\\\\\n\\end{align}","ts":1598869222205,"cs":"gK80WpZc1LoRLZe3NPFqVw==","size":{"width":225,"height":72}}$
Again Differentiate w.r.t. x:-
${"font":{"size":8,"family":"Arial","color":"#000000"},"type":"align","code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(Ame^{mx}+Bne^{nx}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={Am^{2}e^{mx}+Bn^{2}e^{nx}}\t\n\\end{align}","id":"2-1-0-1-0-1-1-1-1-1-1-1-1-0","ts":1598869967746,"cs":"GKZcYrZ8+X/s7MIr3OhwCQ==","size":{"width":164,"height":64}}$
Taking L.H.S:-
${"type":"align","id":"8","code":"\\begin{align}\n{}&={\\frac{d^{2}y}{dx^{2}}-\\left(m+n\\right)\\diff{y}{x}+mny}\\\\\n{}&={Am^{2}e^{mx}+Bn^{2}e^{nx}-\\left(m+n\\right)\\left(Ame^{mx}+Bne^{nx}\\right)+mn\\left(Ae^{mx}+Be^{nx}\\right)}\\\\\n{}&={Am^{2}e^{mx}+Bn^{2}e^{nx}-Am^{2}e^{mx}-Bn^{2}e^{nx}-Amne^{mx}-Bmne^{nx}+Amne^{mx}+Bmne^{nx}}\\\\\n{}&={0}\t\n\\end{align*}","font":{"family":"Arial","size":8,"color":"#000000"},"ts":1598870903458,"cs":"c+WYJa8UgmDWMXuzI/i+Fw==","size":{"width":514,"height":86}}$

15. If y = 500e7x + 600e-7x , show that

${"id":"7-1-0","font":{"family":"Arial","size":12,"color":"#000000"},"code":"$\\frac{d^{2}y}{dx^{2}}=49y$","type":"$","ts":1598871300228,"cs":"Hfh7W0sc42SK3FdYILZZAw==","size":{"width":88,"height":29}}$
Solun:- Given y = 500e7x + 600e-7x
Differentiate w.r.t. x:-
${"code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(500e^{7x }+ 600e^{-7x}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={\\left(3500e^{7x}-4200e^{-7x}\\right).....\\left(1\\right)}\t\n\\end{align}","type":"align","font":{"size":10,"family":"Arial","color":"#000000"},"id":"1-1-0-1-1-1-2-1-1-1-1-1-0-0","ts":1598871598999,"cs":"BKUo2wrcxtMi9yRNkM3N+w==","size":{"width":248,"height":74}}$
Again Differentiate w.r.t. x:-
${"type":"align","code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(3500e^{7x}-4200e^{-7x}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={3500\\times 7.e^{7x}+4200\\times 7.e^{-7x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={7\\left(3500.e^{7x}+4200.e^{-7x}\\right)}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={49\\left(500.e^{7x}+600.e^{-7x}\\right)}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={49y}\t\n\\end{align}","id":"2-1-0-1-0-1-1-1-1-1-1-1-1-1-0-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1598872453866,"cs":"2FeFrKq775R4LfVUUtUlSA==","size":{"width":246,"height":198}}$ (Hence Proved....)

16. If ey(x + 1) = 1, show that

${"font":{"family":"Arial","size":12,"color":"#000000"},"id":"7-1-1","code":"$\\frac{d^{2}y}{dx^{2}}=\\left(\\diff{y}{x}\\right)^{2}$","type":"$","ts":1598872659348,"cs":"Qz5bmWTfm7y8wNHuqBrRDQ==","size":{"width":114,"height":40}}$
Solun:- Given ey(x + 1) = 1
Differentiate w.r.t. x:-
${"type":"align","font":{"family":"Arial","color":"#000000","size":10},"id":"1-1-0-1-1-1-2-1-1-1-1-1-1","code":"\\begin{align}\n{e^{y}\\diff{\\left(x+1\\right)}{x}+\\left(x+1\\right)\\diff{\\left(e^{y}\\right)}{x}}&={0}\\\\\n{e^{y}\\times1+\\left(x+1\\right).e^{y}\\diff{y}{x}}&={0}\\\\\n\\end{align}","ts":1598872880956,"cs":"Q+nKgaukGJ/gBZj48kLrCA==","size":{"width":216,"height":72}}$
${"code":"$\\diff{y}{x}=\\frac{-1}{x+1}$","type":"$","id":"9","font":{"family":"Arial","size":12,"color":"#000000"},"ts":1598873138672,"cs":"frqkUFzxayQLINrJMwUhmw==","size":{"width":88,"height":28}}$
Again Differentiate w.r.t. x:-
${"id":"2-1-0-1-0-1-1-1-1-1-1-1-1-1-1","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align}\n{\\frac{d^{2}y}{dx^{2}}}&={\\diff{\\left(\\frac{-1}{x+1}\\right)}{x}}\\\\\n{\\frac{d^{2}y}{dx^{2}}}&={\\left(\\frac{1}{x+1}\\right)^{2}}\t\n\\end{align}","type":"align","ts":1598873284224,"cs":"8s9hytGNZBBp4LQtBvbG8g==","size":{"width":124,"height":84}}$
${"id":"10","type":"$","code":"$\\frac{d^{2}y}{dx^{2}}=\\left(\\diff{y}{x}\\right)^{2}$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1598873423020,"cs":"Pba4u+QeAFyNtAR/GD8Afg==","size":{"width":114,"height":40}}$ (Hence Proved....)

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

Solun:- Given y = (tan-1x)2
Differentiate w.r.t. x:-
${"type":"align","id":"1-1-0-1-1-1-2-1-1-1-1-1-0-1","code":"\\begin{align}\n{\\diff{y}{x}}&={\\diff{\\left(\\left(\\tan^{-1}x\\right)^{2}\\right)}{x}}\\\\\n{\\diff{y}{x}}&={2\\left(\\tan^{-1}x\\right)\\times\\frac{1}{1+x^{2}}}\\\\\n{\\diff{y}{x}}&={\\frac{2\\left(\\tan^{-1}x\\right)}{1+x^{2}}.....\\left(1\\right)}\t\n\\end{align}","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1598873880511,"cs":"F6RJqQSqqZm7vMVksg+TmA==","size":{"width":186,"height":129}}$
${"id":"11","type":"$","code":"$\\left(1+x^{2}\\right)y_{1}=2\\tan^{-1}x.....\\left(1\\right)$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1598875143146,"cs":"KmPKq7tLzYp6HBUqWJVeQA==","size":{"width":268,"height":24}}$
Again Differentiate w.r.t. x:-
${"id":"2-1-0-1-0-1-1-1-1-1-1-1-1-1-0-1","type":"align","code":"\\begin{align}\n{\\left(1+x^{2}\\right)\\diff{y_{1}}{x}+y_{1}\\diff{\\left(1+x^{2}\\right)}{x}}&={\\diff{\\left(2\\tan^{-1}x\\right)}{x}}\\\\\n{\\left(1+x^{2}\\right)y_{2}+2xy_{1}}&={\\frac{2}{1+x^{2}}}\\\\\n{\\left(1+x^{2}\\right)^{2}y_{2}+2x\\left(1+x^{2}\\right)y_{1}}&={2}\t\n\\end{align}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1598875395293,"cs":"mFGXC8fMYxgisRpoYk701Q==","size":{"width":304,"height":102}}$
(Hence Proved....)

Download PDF of Exercise 5.7
SEE ALSO:-
Notes of continuity and Differentiability
Exercise 5.1
Exercise 5.2
Exercise 5.3
Exercise 5.4
Exercise 5.5
Exercise 5.6

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