If u = tan ^-1 \(\left( \frac { { x }^{ 3 }+{ y }^{ 3 } }{ x-y } \right) \) Prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } \) sin 2u.

Find \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } ,\frac { { \partial }^{ 2 }f }{ { \partial y }^{ 2 } } \)  at x = 2, y = 3 if f(x,y) = 2x² + 3y² - 2xy

Using differential find the approximate value of cos 61; if it is given that sin 60° = 0.86603 and 1⁰ = 0.01745 radians.

If V = log r and r² = x² +y² + z², then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } } \)

If z = f(x - cy) + F (x + cy) where f and F are any two functions and c is a constant, show that \(\frac { { \partial }^{ 2 }z }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }z }{ \partial { y }^{ 2 } } \)

Find \(\frac { \partial w }{ \partial u } ,\frac { \partial w }{ \partial v } \) if w=sin^-1(x,y) where x=u+v,y=u-v

Find the approximate value of \(\sqrt [ 3 ]{ 1.02 } +\sqrt { 1.02 } \)

Find the approximate value of \(\log _{10}\)10.1, it is being given that \(\log _{10} e=0.4343 .\)

Use differential to approximate \((25)^{1 / 3}\)

Using Euler's theorem, prove that \(\mathrm{x} \frac{\hat{\partial} u}{\hat{\partial} x}+\mathrm{y} \frac{\hat{\partial} u}{\partial y}=\frac{1}{2} \tan u\) if \(u=\sin ^{-1}\left(\frac{x-y}{\sqrt{x}+\sqrt{y}}\right)\)

Verify Euler's Theorem for \(f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}}}\)

Suppose that \(\mathrm{Z}=y e^{x^{2}}\) where x = 2t and y = 1- t then find \(\frac{d Z}{d t}\)

If \(w=u^{2} e^{v}\) where \(\mathrm{u}=\frac{x}{y}\) and v = y log x find \(\frac{\partial w}{\partial x} \text { and } \frac{\partial w}{\partial y}\)

 If w = x + 2y + z² and x = cos t, y = sin t, z = t, find \(\frac{d w}{d t}\)

If  \(\mathrm{V}= {\mathrm{z}} \mathrm{e}^{\mathrm{ax}+b \mathrm{y}}\) and z is a homogeneous function of degree n in x and y prove that \(x \frac{\partial V}{\partial x}+\mathbf{y} \frac{\partial V}{\partial y}=(a x+b y+n) V\)