Please note: I am a person studying high school mathematics on my own (well after high school). Furthermore, English is not my first language, which may affect my use of math terminology.
The following has had me confused to no end:
Assume we have the function
$f(x)=(2x+2)^{2}$
if we square the expression we get
$4x^{2}+8x+4$
which, if we derived term for term results in
$f’(x)=8x+8$
Now, the function could be seen as a compounded function, for example $f(g(x))$ where $f(g)=g^{2}$ and $g(x)= 2x+2$
According to the chain rule, the derivative of this compounded function is equal to the outer derivative multiplied by the inner derivative
$f’(g(x)) = f’\times g’ = 2g\times 2 = 2\times 2(2x+2) = 8x+8$
This is all well, but a problem from my book got me confused:
A square has a side x. how fast does the area increase if the side is 12 cm and the side increases by 1,5 cm per minute?
Prior to this, I was introduced to new form of notation of the chain rule
$\dfrac {dA}{dt} = \dfrac {dA}{dx}\times \dfrac {dx}{dt}$
So, the rate of change is equal to $2x\times 1,5$ and if $x$ is set to $12$ then the rate of change is equal to $36 cm^{2}$ per minute.
Here’s my question then: why can’t this be expressed on the first form of the chain rule above? If $x$ is first set to $12$ and the function becomes
$A(t) = (12+1,5t)^{2}$
then the derivative, according to the chain rule is
$36+4,5t$
Why is this way now wrong and the other way correct? What am I missing? Thanks for any help.