If w= log(x²+y²) and x=rcosፀ and y=rsinፀ then, find \(\frac { \partial w }{ \partial r } and\frac { \partial w }{ \partial \theta } \)

If w=xy+z and x=cot, y=sint, z=t then find \(\frac { dw }{ dt } \)

Using linear approximation find \(\sqrt { 0.082 } \)

Find the approximate value of \(\left( \frac { 17 }{ 81 } \right) ^{ \frac { 1 }{ 4 } }\) using linear approximation.

Find the limit for the following if it exists \(\underset { (x-y)\rightarrow \left( 1,1 \right) }{ lim } \frac { { 2x }^{ 2 }-xy-{ y }^{ 2 } }{ { x }^{ 2 }-{ y }^{ 2 } } \) 

Find the limit for the following if it exists \(\underset { (x,y)\rightarrow \left( 0,0 \right) }{ lim } \frac { { x }^{ 2 }{ y }^{ 2 } }{ { x }^{ 4 }+3{ y }^{ 4 } } \)

If w=x²+y² and x=u²-v^2,y=2uv then find \(\frac { \partial w }{ \complement u } and\frac { \partial w }{ \partial v } \)

Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 2,0 \right) }{ lim } \frac { \sqrt { 2x-y-2 } }{ 2x-y-4 } \)

Evaluate : \(\underset { \left( x,y,z \right) \rightarrow \left( -1,0,4 \right) }{ lim } \frac { { x }^{ 2 }-{ ze }^{ zy } }{ 6x+2y-2z } \)

Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 0,0 \right) }{ lim } \frac { { x }^{ 2 }-xy }{ \sqrt { x } -\sqrt { y } } \)

If w=x+2y+z² and x=cos t,y=sint,z=t then find \(\frac { dw }{ dt } \) 

If u(x, y) = \(x^{4}+y^{3}+3 x^{2} y^{2}+3 x^{2} y\) then verify \(\frac{\partial^{2} u}{\partial x \partial y}=\frac{\partial^{2} u}{\partial y \partial x}\)

If u = log (tan x + tan y + tan z), prove that \(\sum \sin 2 x \frac{\partial u}{\partial x}=2\)

If U = (x - y) (y -z) (z- x) then show that \(U_{x}+U_{y}+U_{z}=0\)

Using Euler's Theorem prove the following.
(i) If u = \(\tan ^{-1}\left(\frac{x^{3}+y^{3}}{x-y}\right)\). Prove that \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=\sin 2 u\)
(ii) \(\mathrm{u}=\mathrm{x y}^{2} \sin (x / y)\) Show that \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=3 u\)
(iii) If \(u=\sqrt{x^{2}+y^{2}}\) show that \(\mathbf{x} \frac{\partial u}{\partial x}+\mathbf{y} \frac{\partial u}{\partial y}=u\)
(iv) If u = \(\mathbf{u}=e^{(x / y)} \sin (x / y)+e^{(y / x)} \cos (y / x)\) Show that x \(\frac{\hat{c} u}{\partial x}+\mathbf{y} \frac{\partial u}{\partial y}=0\)