Partial Derivatives of z: Find x,y in z(x, y)

In summary, when finding the two first-order partial derivatives of z with respect to x and y, one must use the product rule and chain rule appropriately. The correct derivatives are dz/dx = yze^xy and dz/dy = xze^xy + z.
  • #1
Scottadams92
1
0
Find the two first-order partial derivatives of z with respect to x and y
when z = z(x, y) is defined implicitly by

z*(e^xy+y)+z^3=1.


I started by multiplying the brackets out to give; ze^xy + zy + z^3 - 1 = 0

i then differentiated each side implicitly and got;
dz/dx = yze^xy
and
dz/dy = xze^xy + z

I'm happy with dz/dx but I've gone wrong on the dz/dy and i don't know where.
 
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  • #2
Scottadams92 said:
Find the two first-order partial derivatives of z with respect to x and y
when z = z(x, y) is defined implicitly by

z*(e^xy+y)+z^3=1.


I started by multiplying the brackets out to give; ze^xy + zy + z^3 - 1 = 0

i then differentiated each side implicitly and got;
dz/dx = yze^xy
and
dz/dy = xze^xy + z

I'm happy with dz/dx but I've gone wrong on the dz/dy and i don't know where.

Show us what you got for ## \frac{\partial z}{\partial y}##

I suspect that when you differentiated zy you neglected to use the product rule, or when you differentiated z3, you didn't use the chain rule.
 

FAQ: Partial Derivatives of z: Find x,y in z(x, y)

1. What is a partial derivative?

A partial derivative is a mathematical concept used to describe the rate of change of a multivariable function with respect to one of its variables, while holding all other variables constant. It is denoted by ∂ and can be thought of as a slope or tangent line on a surface.

2. What is the purpose of finding x and y in z(x, y)?

By finding x and y in z(x, y), we are able to determine how the output of a function, z, changes as its input variables, x and y, change. This allows us to analyze how small changes in x and y affect the overall behavior of the function.

3. How do you find the partial derivatives of z(x, y)?

To find the partial derivatives of z(x, y), we take the derivative of the function with respect to each variable separately, treating all other variables as constants. This results in two partial derivatives: ∂z/∂x and ∂z/∂y.

4. What is the difference between a partial derivative and a regular derivative?

A partial derivative only considers the rate of change of a function with respect to one of its variables, while holding all other variables constant. On the other hand, a regular derivative considers the overall rate of change of a function with respect to its independent variable.

5. Can partial derivatives be applied to any type of function?

Yes, partial derivatives can be applied to any multivariable function. They are particularly useful in fields such as physics, economics, and engineering, where variables often depend on multiple factors.

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