If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) is equal to

If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to

If f(x, y, z) = sin (xy) + sin (yz) + sin (zx) then f_xx is _____________

If y = sin x and x changes from \(\frac{\pi}{2}\) to ㅠ the approximate change in y is ..............

If u = log \(\left( \frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } \right) \) then
(1) u is a homogeneous function
(2) \(x\frac { { \partial }u }{ \partial { x } } +y\frac { { \partial }u }{ { \partial y } } \) = 0
(3) \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } \) is a homogeneous function
(4) \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } \) is a homogeneous function of  degree 0.

Find a linear approximation for the following functions at the indicated points.
f(x) = x³ - 5x + 12, x₀ = 2

Find differential dy for each of the following function
y = (3 + sin(2x)) ^2/3 

Let g(x, y) = \(\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } } \) for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}\) g(x, y) = 0 along every line y = mx, m ∈ R

If f (x, y) = 2x³ - 11x²y + 3y³, prove that \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } =3f\)

IF u(x, y) = x² + 3xy + y², x, y, ∈ R, find tha linear appraoximation for u at (2, 1) 

The trunk of a tree has diameter 30 cm. During the following year, the circumference grew 6cm.
(i) Approximately, how much did the tree's diameter grow?
(ii) What is the percentage increase in area of the tree's cross-section?

prove that g(x, y) = x log\(\left( \frac { y }{ x } \right) \) is homogeneous; what is the degree? Verify Euler's Theorem for g.

Using differentials find the approximate value of tan 46° if it is given that 1⁰ = 0.01745 radians

Assuming log₁₀e = 0.4343, find an approximate value of log₁₀ 1003

For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = log (5x + 3y)

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