What does it mean to be be an inside or outside function?
for example:
tell which is the inside function and which is the outside:
f(x)=sin3x
The idea is that
df (g(x) )/dx = df/dg * dg/dx
where f is the "outside function and g is the "inside" function
You do the outside function derivative first, then the inside function
so for your example we have
f ( g(x) ) = sin (g(x)) = sin ( 3x )
f (g) = sin g outside
g = 3x inside
so in your case
df/dg = cos g outside = cos 3x
dg/dx = 3 inside
so df/dx = cos g * 3
or 3 cos 3x
When we have a composite function, it means that one function is being applied to the result of another function. The function being applied last is called the outside function, while the function being applied first is called the inside function.
In the example you provided, f(x) = sin(3x), we can see that the sine function is being applied to the result of 3x. So, in this case, the inside function is 3x, and the outside function is sin.
To determine the inside and outside functions, we can think about the order of operations. The inside function is the one that you would evaluate first before applying the outside function. In f(x) = sin(3x), you would evaluate 3x first, and then apply the sine function to the result.
So, in this case:
Inside function: 3x
Outside function: sin