- #1
bobsmith76
- 336
- 0
Homework Statement
I don't understand why
∂f/∂x = xy = y
whereas
∂f/∂x = x2 + y2 = 2x
Why does the y disappear in the second but not in the first?
Confusion with Partial Derivatives: Why does y disappear? | Explained
- Thread starter bobsmith76
- Start date
In summary, when taking partial derivatives, y disappears because we are only looking at the change in one variable while holding all others constant. Partial derivatives can still be used even if y is not a function of x. A partial derivative being equal to zero means that the function is not changing in the direction of that variable. The variable to take the partial derivative with respect to is usually specified or the one we are trying to analyze. The chain rule can be used when taking partial derivatives, but it results in an additional term denoted by ∂.
I don't understand why
∂f/∂x = xy = y
whereas
∂f/∂x = x2 + y2 = 2x
Why does the y disappear in the second but not in the first?
- #2
tiny-tim
Science Advisor
Homework Helper
- 25,839
- 258
hi bobsmith76! 
you mean? …
in each case, y is treated as a constant
in the first, it's multiplied, so it stays; in the second, it's on its own, so its derivative is zero
you mean? …
∂/∂x (xy) = y
∂/∂x (x2 + y2) = 2x
∂/∂x (x2 + y2) = 2x
in each case, y is treated as a constant
in the first, it's multiplied, so it stays; in the second, it's on its own, so its derivative is zero
∂/∂x means differentiating wrt x while keeping all other variables constant
1. Why does the variable y disappear when taking partial derivatives?
When taking partial derivatives, we are only considering the change in one variable while holding all other variables constant. This means that we are ignoring the impact of any other variables on the function, including y. Therefore, when taking the partial derivative with respect to x, for example, y is treated as a constant and therefore disappears from the equation.
2. Can we still use partial derivatives if y is not a function of x?
Yes, partial derivatives can still be used even if y is not a function of x. This is because when taking the partial derivative with respect to x, we are still only considering the change in x while holding all other variables constant. So even if y is not explicitly a function of x, it is still treated as a constant and disappears from the equation.
3. What does it mean when a partial derivative is equal to zero?
If a partial derivative is equal to zero, it means that the function is not changing in the direction of that variable. In other words, the slope of the function in that direction is flat. This could also indicate a maximum or minimum point in the function, as the slope would change from positive to negative or vice versa at that point.
4. How do you know which variable to take the partial derivative with respect to?
The variable to take the partial derivative with respect to is usually specified in the problem or context. If not, it is typically the variable that we are trying to analyze or understand the relationship of with the other variables in the function. For example, if we are trying to find the rate of change of a function with respect to time, we would take the partial derivative with respect to time.
5. Can you use the chain rule when taking partial derivatives?
Yes, the chain rule can be used when taking partial derivatives. However, it is important to note that the chain rule will result in an additional term when taking partial derivatives, as we are only considering the change in one variable at a time. This additional term is known as the "partial derivative of the inner function" and is denoted by the symbol ∂.
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