Math Problems Vault

Problem 1111

Find $y''$ , given: $y=x\sqrt{1+x^{2}}$ .

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Problem 1112

Find $y''$ , given: $y=\dfrac {x}{\sqrt{1-x^{2}}}$ .

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Problem 1113

Find $y''$ , given: $y=e^{-x^{2}}$ .

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Problem 1114

Find $y''$ , given: $y=\tg x$ .

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Problem 1115

Find $y''$ , given: $y=(1+x^{2})\arctg x$ .

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Problem 1116

Find $y''$ , given: $y=\dfrac {\arcsin x}{\sqrt{1-x^{2}}}$ .

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Problem 1117

Find $y''$ , given: $y=x\ln x$ .

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Problem 1118

Find $y''$ , given: $y=\ln f(x)$ .

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Problem 1119

Find $y''$ , given: $y=x\left[\sin (\ln x)+\cos (\ln x)\right]$ .

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Problem 1120

Let $y=e^{\sin x}\cos (\sin x)$ , find $y(0)$ , $y'(0)$ and $y''(0)$ .

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Problem 1121

Let $u=\varphi (x)$ and $v=\psi (x)$ be twice differentiable functions. Find $y''$ , given: $y=u^{2}$ .

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Problem 1122

Let $u=\varphi (x)$ and $v=\psi (x)$ be twice differentiable functions. Find $y''$ , given: $y=\ln \dfrac {u}{v}$ .

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Problem 1123

Let $u=\varphi (x)$ and $v=\psi (x)$ be twice differentiable functions. Find $y''$ , given: $y=\sqrt{u^{2}+v^{2}}$ .

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Problem 1124

Let $u=\varphi (x)$ and $v=\psi (x)$ be twice differentiable functions. Find $y''$ , given: $y=u^{v} \ (u>0)$ .

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Problem 1125

Let $f(x)$ be a thrice differentiable function. Find $y''$ and $y'''$ , given: $y=f(x^{2})$ .

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Problem 1126

Let $f(x)$ be a thrice differentiable function. Find $y''$ and $y'''$ , given: $y=f\left(\dfrac {1}{x}\right)$ .

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Problem 1127

Let $f(x)$ be a thrice differentiable function. Find $y''$ and $y'''$ , given: $y=f(e^{x})$ .

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Problem 1128

Let $f(x)$ be a thrice differentiable function. Find $y''$ and $y'''$ , given: $y=f(\ln x)$ .

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Problem 1129

Let $f(x)$ be a thrice differentiable function. Find $y''$ and $y'''$ , given: $y=f[\varphi (x)]$ , where $\varphi (x)$ is a function differentiable as many times as needed.

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Problem 1130

(a)(b)

Find $d^{2}y$ for the function $y=e^{x}$ in the following two cases:

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(a)

$x$ is the independent variable;

(b)

$x$ is an intermediate variable.

Problem 1131

Treating $x$ as the independent variable, find $d^{2}y$ , given: $y=\sqrt{1+x^{2}}$ .

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Problem 1132

Treating $x$ as the independent variable, find $d^{2}y$ , given: $y=\dfrac {\ln x}{x}$ .

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Problem 1133

Treating $x$ as the independent variable, find $d^{2}y$ , given: $y=x^{x}$ .

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Problem 1134

Let $u$ and $v$ be twice differentiable functions of the variable $x$ . Find $d^{2}y$ , given: $y=uv$ .

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Problem 1135

Let $u$ and $v$ be twice differentiable functions of the variable $x$ . Find $d^{2}y$ , given: $y=\dfrac {u}{v}$ .

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Problem 1136

Let $u$ and $v$ be twice differentiable functions of the variable $x$ . Find $d^{2}y$ , given: $y=u^{m}v^{n}$ ( $m$ and $n$ are constants).

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Problem 1137

Let $u$ and $v$ be twice differentiable functions of the variable $x$ . Find $d^{2}y$ , given: $y=a^{u} \ (a>0)$ .

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Problem 1138

Let $u$ and $v$ be twice differentiable functions of the variable $x$ . Find $d^{2}y$ , given: $y=\ln \sqrt{u^{2}+v^{2}}$ .

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Problem 1139

Let $u$ and $v$ be twice differentiable functions of the variable $x$ . Find $d^{2}y$ , given: $y=\arctg \dfrac {u}{v}$ .

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Problem 1140

Find the derivatives $y'_{x}$ , $y''_{x^{2}}$ , $y'''_{x^{3}}$ of the function $y=y(x)$ given parametrically: $x=2t-t^{2}$ , $y=3t-t^{3}$ .

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Problem 1141

Find the derivatives $y'_{x}$ , $y''_{x^{2}}$ , $y'''_{x^{3}}$ of the function $y=y(x)$ given parametrically: $x=a\cos t$ , $y=a\sin t$ .

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Problem 1142

Find the derivatives $y'_{x}$ , $y''_{x^{2}}$ , $y'''_{x^{3}}$ of the function $y=y(x)$ given parametrically: $x=a(t-\sin t)$ , $y=a(1-\cos t)$ .

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Problem 1143

Find the derivatives $y'_{x}$ , $y''_{x^{2}}$ , $y'''_{x^{3}}$ of the function $y=y(x)$ given parametrically: $x=e^{t}\cos t$ , $y=e^{t}\sin t$ .

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Problem 1144

Find the derivatives $y'_{x}$ , $y''_{x^{2}}$ , $y'''_{x^{3}}$ of the function $y=y(x)$ given parametrically: $x=f'(t)$ , $y=tf'(t)-f(t)$ .

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Problem 1145

Let the function $y=f(x)$ be differentiable a certain number of times. Find the derivatives $x'$ , $x''$ , $x'''$ , $x^{(4)}$ of the inverse function $x=f^{-1}(y)$ (assuming all these derivatives exist).

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Problem 1146

Find $y'_{x}$ , $y''_{x^{2}}$ and $y'''_{x^{3}}$ of the function $y=y(x)$ given implicitly: $x^{2}+y^{2}=25$ . What are $y'$ , $y''$ , and $y'''$ at the point $M(3,4)$ ?

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Problem 1147

Find $y'_{x}$ , $y''_{x^{2}}$ and $y'''_{x^{3}}$ of the function $y=y(x)$ given implicitly: $y^{2}=2px$ .

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Problem 1148

Find $y'_{x}$ , $y''_{x^{2}}$ and $y'''_{x^{3}}$ of the function $y=y(x)$ given implicitly: $x^{2}-xy+y^{2}=1$ .

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Problem 1149

Find $y'_{x}$ and $y''_{x^{2}}$ , given: $y^{2}+2\ln y=x^{4}$ .

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Problem 1150

Find $y'_{x}$ and $y''_{x^{2}}$ , given: $\sqrt{x^{2}+y^{2}}=ae^{\arctg \frac{y}{x}} \ (a>0)$ .

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Problem 1151

Suppose the function $f(x)$ is defined and twice differentiable for $x \leqslant x_{0}$ . How should the coefficients $a,b$ and $c$ be chosen so that the function

F(x)= \begin{cases} f(x), & \text{if } x \leqslant x_{0}, \\ a(x-x_{0})^{2}+b(x-x_{0})+c, & \text{if } x>x_{0} \end{cases}

is twice differentiable?

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Problem 1152

A point moves in a straight line according to the law

s=10+20t-5t^{2} .

Find the velocity and acceleration of the motion. What are the velocity and acceleration at the instant $t=2$ ?

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Problem 1153

A point $M(x,y)$ moves uniformly along the circle $x^{2}+y^{2}=a^{2}$ , completing one revolution every $T$ seconds. Find the velocity $v$ and acceleration $j$ of the projection of point $M$ onto the $Ox$ axis, given that at $t=0$ the position of the point is $M_{0}(a,0)$ .

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Problem 1154

A particle $M(x,y)$ is thrown in the vertical plane $Oxy$ with initial velocity $v_{0}$ in a direction making angle $\alpha$ with the horizontal. Set up the equations of motion (neglecting air resistance) and compute the magnitudes of the velocity $v$ and acceleration $j$ , and the trajectory of the motion. What are the maximum height and range?

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Problem 1155

A point moves according to the equations $x=4\sin \omega t-3\cos \omega t$ , $y=3\sin \omega t+4\cos \omega t$ ( $\omega$ is a constant). Find the trajectory of the motion and the magnitudes of the velocity and acceleration.

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Problem 1156

Find the derivative of the indicated order: $y=x(2x-1)^{2}(x+3)^{3}$ ; find $y^{(6)}$ and $y^{(7)}$ .

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Problem 1157

Find the derivative of the indicated order: $y=\dfrac {a}{x^{m}}$ , find $y'''$ .

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Problem 1158

Find the derivative of the indicated order: $y=\sqrt{x}$ , find $y^{(10)}$ .

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Problem 1159

Find the derivative of the indicated order: $y=\dfrac {x^{2}}{1-x}$ , find $y^{(8)}$ .

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Problem 1160

Find the derivative of the indicated order: $y=\dfrac {1+x}{\sqrt{1-x}}$ , find $y^{(100)}$ .

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Problem 1161

Find the derivative of the indicated order: $y=x^{2}e^{2x}$ , find $y^{(20)}$ .

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Problem 1162

Find the derivative of the indicated order: $y=\dfrac {e^{x}}{x}$ , find $y^{(10)}$ .

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Problem 1163

Find the derivative of the indicated order: $y=x\ln x$ , find $y^{(5)}$ .

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Problem 1164

Find the derivative of the indicated order: $y=\dfrac {\ln x}{x}$ , find $y^{(5)}$ .

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Problem 1165

Find the derivative of the indicated order: $y=x^{2}\sin 2x$ , find $y^{(50)}$ .

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Problem 1166

Find the derivative of the indicated order: $y=\dfrac {\cos 3x}{\sqrt[3]{1-3x}}$ , find $y'''$ .

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Problem 1167

Find the derivative of the indicated order: $y=\sin x\sin 2x\sin 3x$ , find $y^{(10)}$ .

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Problem 1168

Find the derivative of the indicated order: $y=x\sh x$ , find $y^{(100)}$ .

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Problem 1169

Find the derivative of the indicated order: $y=e^{x}\cos x$ , find $y^{(4)}$ .

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Problem 1170

Find the derivative of the indicated order: $y=\sin^{2}x\ln x$ , find $y^{(6)}$ .

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Problem 1171

In each of the following examples, treating $x$ as the independent variable, find the differential of the indicated order: $y=x^{5}$ , find $d^{5}y$ .

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Problem 1172

In each of the following examples, treating $x$ as the independent variable, find the differential of the indicated order: $y=\dfrac {1}{\sqrt{x}}$ , find $d^{3}y$ .

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Problem 1173

In each of the following examples, treating $x$ as the independent variable, find the differential of the indicated order: $y=x\cos 2x$ , find $d^{10}y$ .

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Problem 1174

In each of the following examples, treating $x$ as the independent variable, find the differential of the indicated order: $y=e^{x}\ln x$ , find $d^{4}y$ .

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Problem 1175

In each of the following examples, treating $x$ as the independent variable, find the differential of the indicated order: $y=\cos x \cdot \ch x$ , find $d^{6}y$ .

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Problem 1176

Let $u$ be a function of $x$ that is differentiable a sufficient number of times; in each of the following examples find the differential of the indicated order: $y=u^{2}$ , find $d^{10}y$ .

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Problem 1177

Let $u$ be a function of $x$ that is differentiable a sufficient number of times; in each of the following examples find the differential of the indicated order: $y=e^{u}$ , find $d^{4}y$ .

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Problem 1178

Let $u$ be a function of $x$ that is differentiable a sufficient number of times; in each of the following examples find the differential of the indicated order: $y=\ln u$ , find $d^{3}y$ .

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Problem 1179

Treating $x$ as a function of some independent variable, find $d^{2}y,d^{3}y$ and $d^{4}y$ for the function $y=f(x)$ .

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Problem 1180

Express the second and third derivatives $y''$ and $y'''$ of the function $y=f(x)$ in terms of successive differentials of the variables $x$ and $y$ , without assuming $x$ to be the independent variable.

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Problem 1181

Prove: the function

y=C_{1}\cos x+C_{2}\sin x,

where $C_{1}$ and $C_{2}$ are arbitrary constants, satisfies the equation

y''+y=0 .

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Problem 1182

Prove: the function

y=C_{1}\ch x+C_{2}\sh x,

where $C_{1}$ and $C_{2}$ are arbitrary constants, satisfies the equation

y''-y=0 .

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Problem 1183

Prove: the function

y=C_{1}e^{\lambda _{1}x}+C_{2}e^{\lambda _{2}x},

where $C_{1}$ and $C_{2}$ are arbitrary constants and $\lambda_{1}$ and $\lambda_{2}$ are constants, satisfies the equation

y''-(\lambda _{1}+\lambda _{2})y'+\lambda _{1}\lambda _{2}y=0 .

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Problem 1184

Prove: the function

y=x^{n}\left[C_{1}\cos (\ln x)+C_{2}\sin (\ln x)\right],

where $C_{1}$ and $C_{2}$ are arbitrary constants and $n$ is a constant, satisfies the equation:

x^{2}y''+(1-2n)xy'+(1+n^{2})y=0 .

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Problem 1185

Prove: the function

\begin{align} y & =e^{-\frac{x}{\sqrt{2}}}\left(C_{1}\cos \frac{x}{\sqrt{2}}+C_{2}\sin \frac{x}{\sqrt{2}}\right) \\ & \quad +e^{-\frac{x}{\sqrt{2}}}\left(C_{3}\cos \frac{x}{\sqrt{2}}+C_{4}\sin \frac{x}{\sqrt{2}}\right), \end{align}

where $C_{1},C_{2},C_{3}$ and $C_{4}$ are arbitrary constants, satisfies the equation

y^{(4)}+y=0 .

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Problem 1186

Prove: if the function $f(x)$ has a derivative of order $n$ , then

\left[f(ax+b)\right]^{(n)}=a^{n}f^{(n)}(ax+b) .

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Problem 1187

If $P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}$ , find $P^{(n)}(x)$ .

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Problem 1188

Find $y^{(n)}$ , given: $y=\dfrac {ax+b}{cx+d}$ .

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Problem 1189

Find $y^{(n)}$ , given: $y=\dfrac {1}{x(1-x)}$ .

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Problem 1190

Find $y^{(n)}$ , given: $y=\dfrac {1}{x^{2}-3x+2}$ .

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Problem 1191

Find $y^{(n)}$ , given: $y=\dfrac {1}{\sqrt{1-2x}}$ .

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Problem 1192

Find $y^{(n)}$ , given: $y=\dfrac {x}{\sqrt[3]{1+x}}$ .

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Problem 1193

Find $y^{(n)}$ , given: $y=\sin^{2}x$ .

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Problem 1194

Find $y^{(n)}$ , given: $y=\cos^{2}x$ .

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Problem 1195

Find $y^{(n)}$ , given: $y=\sin^{3}x$ .

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Problem 1196

Find $y^{(n)}$ , given: $y=\cos^{3}x$ .

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Problem 1197

Find $y^{(n)}$ , given: $y=\sin ax\sin bx$ .

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Problem 1198

Find $y^{(n)}$ , given: $y=\cos ax\cos bx$ .

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Problem 1199

Find $y^{(n)}$ , given: $y=\sin ax\cos bx$ .

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Problem 1200

Find $y^{(n)}$ , given: $y=\sin^{2}ax\cos bx$ .

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Problem 1201

Find $y^{(n)}$ , given: $y=\sin^{4}x+\cos^{4}x$ .

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Problem 1202

Find $y^{(n)}$ , given: $y=x\cos ax$ .

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Problem 1203

Find $y^{(n)}$ , given: $y=x^{2}\sin ax$ .

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Problem 1204

Find $y^{(n)}$ , given: $y=(x^{2}+2x+2)e^{-x}$ .

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Problem 1205

Find $y^{(n)}$ , given: $y=\dfrac {e^{x}}{x}$ .

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Problem 1206

Find $y^{(n)}$ , given: $y=e^{x}\cos x$ .

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Problem 1207

Find $y^{(n)}$ , given: $y=e^{x}\sin x$ .

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Problem 1208

Find $y^{(n)}$ , given: $y=\ln \dfrac {a+bx}{a-bx}$ .

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Problem 1209

Find $y^{(n)}$ , given: $y=e^{ax}P(x)$ , where $P(x)$ is a polynomial.

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Problem 1210

Find $y^{(n)}$ , given: $y=x\sh x$ .

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Problem 1211

Find $d^{n}y$ , given $y=x^{n}e^{x}$ .

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Problem 1212

Find $d^{n}y$ , given $y=\dfrac {\ln x}{x}$ .

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Problem 1213

(a)(b)

Prove the identities:

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(a)

$\left[e^{ax}\sin (bx+c)\right]^{(n)}$

=e^{ax}(a^{2}+b^{2})^{\frac{n}{2}}\sin (bx+c+n\varphi )

(b)

$\left[e^{ax}\cos (bx+c)\right]^{(n)}$

=e^{ax}(a^{2}+b^{2})^{\frac{n}{2}}\cos (bx+c+n\varphi ),

where $\sin \varphi =\dfrac {b}{\sqrt{a^{2}+b^{2}}}$ and $\cos \varphi =\dfrac {a}{\sqrt{a^{2}+b^{2}}}$ .

Problem 1214

(a)(b)(c)(d)

Find $y^{(n)}$ , given:

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(a)

$y=\ch ax\cos bx$ ;

(b)

$y=\ch ax\sin bx$ ;

(c)

$y=\sh ax\cos bx$ ;

(d)

$y=\sh ax\sin bx$ .

Problem 1215

Express the function

f(x)=\sin ^{2p}x,

where $p$ is a natural number, as a trigonometric polynomial:

f(x)=\sum _{k=0}^{p}A_{k}\cos 2kx,

in order to find $f^{(n)}(x)$ .

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Problem 1216

(a)(b)(c)

Given:

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(a)

$f(x)=\sin^{2p+1}x$ ;

(b)

$f(x)=\cos^{2p}x$ ;

(c)

$f(x)=\cos^{2p+1}x$ ,

where $p$ is a positive integer, find $f^{(n)}(x)$ .

Problem 1217

Using the identity

\frac{1}{x^{2}+1}=\frac{1}{2i}\left(\frac{1}{x-i}-\frac{1}{x+i}\right)

prove that

\left(\frac{1}{x^{2}+1}\right)^{(n)}=\frac{(-1)^{n}n!}{(1+x^{2})^{\frac{n+1}{2}}}\sin \left[(n+1)\arcctg x\right] .

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Problem 1218

Find the $n$ -th order derivative of the function $f(x)=\arctg x$ .

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Problem 1219

(a)(b)

Find $f^{(n)}(0)$ , given:

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(a)

$f(x)=\dfrac {1}{(1-2x)(1+x)}$ ;

(b)

$f(x)=\dfrac {x}{\sqrt{1-x}}$ .

Problem 1220

(a)(b)(c)

Given:

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(a)

$f(x)=x^{2}e^{ax}$ ;

(b)

$f(x)=\arctg x$ ;

(c)

$f(x)=\arcsin x$ .

Find $f^{(n)}(0)$ .

Problem 1221

(a)(b)

Given:

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(a)

$f(x)=\cos (m\arcsin x)$ ;

(b)

$f(x)=\sin (m\arcsin x)$ .

Problem 1222

(a)(b)

Given:

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(a)

$f(x)=(\arctg x)^{2}$ ;

(b)

$f(x)=(\arcsin x)^{2}$ .

Problem 1223

Given

f(x)=(x-a)^{n}\varphi (x),

where the function $\varphi (x)$ has a continuous derivative of order $(n-1)$ in a neighborhood of the point $a$ , find $f^{(n)}(a)$ .

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Problem 1224

Prove that the function

f(x)= \begin{cases} x^{2n}\sin \dfrac {1}{x}, & \text{if } x \neq 0, \\ 0, & \text{if } x=0 \end{cases}

( $n$ a natural number) has, at the point $x=0$ , derivatives up to order $n$ , but no derivative of order $(n+1)$ .

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Problem 1225

Prove that the function

f(x)= \begin{cases} e^{-\frac{1}{x^{2}}}, & \text{if } x \neq 0, \\ 0, & \text{if } x=0, \end{cases}

is infinitely differentiable at $x=0$ . Sketch the graph of this function.

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Problem 1226

Prove that the Chebyshev polynomials

T_{m}(x)=\frac{1}{2^{m-1}}\cos (m\arccos x) \ (m=0,1,2,\cdots )

satisfy the equation

(1-x^{2})T''_{m}(x)-xT'_{m}(x)+m^{2}T_{m}(x)=0 .

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Problem 1227

Prove that the Legendre polynomials

P_{m}(x)=\frac{1}{2^{m}m!}\left[(x^{2}-1)^{m}\right]^{(m)} \ (m=0,1,2,\cdots )

satisfy the equation

(1-x^{2})P''_{m}(x)-2xP'_{m}(x)+m(m+1)P_{m}(x)=0 .

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Problem 1228

The Chebyshev--Laguerre polynomials are defined as follows:

L_{m}(x)=e^{x}(x^{m}e^{-x})^{(m)} \ (m=0,1,2,\cdots ) .

Find an explicit expression for the polynomial $L_{m}(x)$ .

Prove that $L_{m}(x)$ satisfies the equation

xL''_{m}(x)+(1-x)L'_{m}(x)+mL_{m}(x)=0 .

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Problem 1229

Let $y=f(u)$ and $u=\varphi (x)$ , where $f(u)$ and $\varphi (x)$ are functions differentiable $n$ times.

Prove that $\dfrac {d^{n}y}{dx^{n}}=\displaystyle \sum_{k=1}^{n}A_{k}(x)f^{(k)}(u)$ , where the coefficients $A_{k}(x) \ (k=0,1,\cdots ,n)$ do not depend on the function $f(u)$ .

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Problem 1230

Prove that for the $n$ -th derivative of the composite function $y=f(x^{2})$ , the following formula holds

\begin{align} \frac{d^{n}y}{dx^{n}} & =(2x)^{n}f^{(n)}(x^{2})-\frac{n(n-1)}{1!}(2x)^{n-2}f^{(n-1)}(x^{2}) \\ & \quad +\frac{n(n-1)(n-2)(n-3)}{2!}(2x)^{n-4}f^{(n-2)}(x^{2}) \\ & \quad +\cdots . \end{align}

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Problem 1231

The Chebyshev--Hermite polynomials are defined as follows:

H_{m}(x)=(-1)^{m}e^{x^{2}}(e^{-x^{2}})^{(m)} \ (m=0,1,2,\cdots ) .

Find an explicit expression for the polynomial $H_{m}(x)$ .

Prove that $H_{m}(x)$ satisfies the equation

H''_{m}(x)-2xH'_{m}(x)+2mH_{m}(x)=0 .

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Problem 1232

Prove the identity

(x^{n-1}e^{\frac{1}{x}})^{n}=\frac{(-1)^{n}}{x^{n+1}}e^{\frac{1}{x}} .

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Problem 1232.1

[NO TASK]

Problem 1232.2

[NO TASK]

Problem 1233

Let $\dfrac {d}{dx}=D$ denote the differential operator and let

f(D)=\sum _{k=0}^{n}p_{k}(x)D^{k}

be a polynomial in the differential symbol, where $p_{k}(x) \ (k=0,1,\cdots ,n)$ are certain continuous functions of $x$ .

Prove that

f(D)\left\{ e^{\lambda x}u(x)\right\} =e^{\lambda x}f(D+\lambda )u(x),

where $\lambda$ is a constant.

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Problem 1234

Prove that in the equation

\sum _{k=0}^{n}a_{k}x^{k}\frac{d^{k}y}{dx^{k}}=0

if we set

x=e^{t},

where $t$ is the independent variable, then this equation takes the form

\sum _{k=0}^{n}a_{k}D(D-1)\cdots (D-k+1)y=0,

where $D=\dfrac {d}{dt}$ .

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