Question

how does the product rule apply for the derivative of two variables say: z(x,y)= xy

Answer 1

Is this a class where you know partial derivatives?

Answer 2

i'm not sure if i have to apply partial derivatives, i have to find the derivative of r(p) = pd(p) to get the elasticity

Answer 3

Okay, so does the question hint at assuming that y is a function of x?
\[y(x) \]

Answer 4

they use d(p) and r(p) and p so i guess they do

Answer 5

well if you said use product rule for r(p)=p*d(p) then I would not think to use partial derivative
but your other question does require parital derivatives

Answer 6

question says:
if d(p) represents the demand function, then the revenue r is equal to the product of the price p and the demand d(p), r(p)= pd(p). the elasticity of the revenue is denoted by er and the elasticity of the demand by ed.

show that the following relation holds

er= 1+ ed

Answer 7

Ah, okay, I though that was for a different question.
\[ r(p) = pd(p)\]

Apply the dervative operator w.r.t. p \[\frac{d}{dp} \left ( r(p) \right ) =\frac{d}{dp} \left ( p d(p) \right ) \]

From the product rule, we can simplify the rhs to:
\[\frac{d}{dp} \left ( r(p) \right ) =\frac{d}{dp}(p) d+ \frac{d}{dp}(d) p \]

Since p and d are functions of p, we get:

\[\frac{d}{dp} \left ( r(p) \right ) =d+ \frac{dd}{dp}p \]