Question

I'm confused with this partial derivative:
z=ye^x/y . Zy=e^x/y
But when I do Zx I get xe^x/y
and the book says otherwise, can somebody please explain me what I'm doing wrong?

Answer 1

is this
\[ z= y\ e^{\frac{x}{y} } \]?

Answer 2

No, the book apparently used the product rule and it has the following:
-x/y e^x/y + e^x/y

Answer 3

I don't know where they are getting the negative from

Answer 4

what is the original expression ?

Answer 5

z=ye^x/y

Answer 6

is x/y in the exponent i.e. as I posted up above ?

Answer 7

yes

Answer 8

what do you get for \( z_x \) ?

Answer 9

e^x/y

Answer 10

yes, that is what I get.

Answer 11

for Zx yes but the problem is for Zy

Answer 12

for Zy, treat x as a constant
for
y exp( x/y)
we need to use the product rule
y d/dy exp(x/y) + exp(x/y) d/dy y
the 2nd term is just exp(x/y)

the first term is
y * exp(x/y) * d/dy (x/y)

Answer 13

I get the second term as just being e^x/y. that's fine, I"m confused as to where they are getting the negative from

Answer 14

d/dy (x/y) (with x constant ) is
x * d/dy y^(-1)

Answer 15

hold on let me get the stylus so that I can write it out

Answer 16

you don't have to write it out.
you just have to work on the first term
\[ y \frac{d}{dy}(e^\frac{x}{y} ) \]
the derivative of e^u is e^u du
so
\[ y \frac{d}{dy}(e^\frac{x}{y} ) \\
y\ e^\frac{x}{y}\frac{d}{dy}\frac{x}{y}
\]