- #1
bobsmith76
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- 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?
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.
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.
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.
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.
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 ∂.