Let f, g : (a, b)→R be differentiable functions. Show that d(fg) = fdg + gdf

Let g(x) = x² + sin x. Calculate the differential dg.

If the radius of a sphere, with radius 10 cm, has to decrease by 0 1. cm, approximately how much will its volume decrease?

Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)

Let F(x, y) = x³ y + y²x + 7 for all (x, y)∈ R². Calculate \(\frac { \partial F }{ \partial x } \)(-1, 3) and \(\frac { \partial F }{ \partial y } \)(-2, 1).

Let w(x, y) = xy+\(\frac { { e }^{ y } }{ { y }^{ 2 }+1 } \) for all (x, y) ∈ R². Calculate \(\frac { { \partial }^{ 2 }w }{ { \partial y\partial x } } \) and \(\frac { { \partial }^{ 2 }w }{ { \partial x\partial y } } \)

Let (x, y) = e^-2y cos(2x) for all (x, y) ∈ R². Prove that u is a harmonic function in R².

Verify the above theorem for F(x, y) = x² - 2y² + 2xy and x(t) = cos t, y(t) = sin t, t ∈ [0, 2\(\pi\)]

Let g( x, y) = x² - yx + sin(x+y), x(t) = e^3t, y(t) = t², t ∈ R. Find \(\frac { dg }{ dt } \) 

Let g(x, y) = 2y + x², x = 2r -s, y = r²+ 2s, r, s ∊ R. Find \(\frac { \partial g }{ \partial r } ,\frac { \partial g }{ \partial s } \)