I initially assumed I wouldn't get the inner derivatives because I wasn't differentiating with respect to the.. "inner variables". But I can't quite put it down in writing in a way that seems convincing to me, hence the uncertainty.arildno said:You say you're not sure. But you also make the point that you are not differentiating with respect to inner variables.
So, what do you think is the correct answer to your question?
A derivative is a mathematical concept that represents the rate of change of a function at a given point. It is calculated by taking the limit of the change in the function over the change in the input as the change in the input approaches zero.
To determine if a derivative is an inner function, you can use the chain rule, which states that the derivative of an outer function multiplied by the derivative of the inner function is equal to the derivative of the composition of the two functions. If this rule applies, then the derivative is an inner function.
Knowing if a derivative is an inner function is important because it can help us simplify complicated functions and make it easier to calculate the derivative. Inner functions also play a crucial role in many mathematical concepts, such as the chain rule and the inverse function theorem.
Visually, inner functions can be identified as functions within functions. For example, in the function f(x) = e^(2x+5), the inner function is 2x+5. Additionally, if a function has a composite structure, where one function is being applied to the output of another function, the inner function is the function that is being applied to the input of the outer function.
Yes, a derivative can be both an inner and outer function. This is because the chain rule applies to compositions of functions, meaning that a function can have multiple layers of inner and outer functions. In these cases, the derivative would be considered both an inner and outer function.