I'm wondering...With
Why doesn't count as an "inside" function?
Example:
Here we see an example of the chain rule. Derivative of the inside function times the derivative of the outside function...
However, with this example:
There is no derivative of the inside and then outside. It seems to adhere to normal differentiation rules such as:
Why is this? Thanks!
Of course not; functions map a range of inputs to a range of outputs, so therefore require at least a single variable to do so. So for instance e^x is a function that maps x to e^x. e^5 is not a function, it always has constant value. Ln(7x) is a function, but ln(7) is not. The derivative (wrt to x) of e^(ln7x) is as has been pointed out, 7, but the derivate of e^((ln7)x) is ln(7)e^((ln7)x) since it is of the form e^kx where k=ln7 so you can see it does indeed follow normal differentiation laws.
Uuhhh... it depends on what we are talking about... is a single 1 dimensional number a function ? I don't know. In 2 dimensions f(x) = c is a function , it is a horizontal line that takes on all x values exactly once and every x value has only 1 y value and therefore passes the so called 'vertical line test'. Similarly f(x) = ln 7 is a function in 2 (or more) dimensions. For any real number c , one may decide to rewrite this as
In order to have a 'place' to sbstitute for x ... this works out fine for all real x EXCEPT x = 0 where we encounter the 'disagreeable'
Personally, it doesn't bother me too much and i won't throw it away just because there is a problem at x = 0 , but that's me , others may disagree.