Question

Partial derivative problem

Answer 1

I've worked out the partial derivative with respect to x but I have a problem with respect to y.
\[f(x,y) = 4e ^{\frac{ x }{ y }} + \tan ^{-1}(\frac{ y }{ x })\]

Answer 2

What I like to do is rewrite the x/y as xy^-1 and yx^-1 and then this is a little trick that I try. so we need to find a partial derivative in terms of y. I say leave x alone and deal with the y meaning take all of the derivatives of y and leave anything with an x alone. Other way of putting it is treat x like a constant and take the derivatives of y only

Answer 3

I get
\[\frac{ -4xe ^{\frac{ x }{ y }} }{ y ^{2} } + \frac{ 1 }{ x+\frac{ y ^{2} }{ x }}\]
but it's different to the book's answer.

Answer 4

hmmm sometimes the book takes an extra step... like simplifying...let me try drawing it

Answer 5

My bad. The book's one is simplified. I feel so dumb now low. Thanks UsukiDoll