Thanks for your answer!

For the last part, how do I differentiate the equation dS/dA to get your differentiated answer?

S = (8 + π) x² + 640 (12 + π) / [x (8 + π)]
S = (8 + π) x² + [640 (12 + π) / (8 + π)] / x
S = (8 + π) x² + [640 (12 + π) / (8 + π)] times x^-1

In our expression, the (8 + π) and the [640 (12 + π) / (8 + π)] are just constant multipliers to the x² and the 1/x respectively, so we perform the usual basic differentiation rules accordingly, that is,

dS/dx
= (8 + π) (2) (x) + [640 (12 + π) / (8 + π)] times (-1) x^-2
= 2 (8 + π) (x) - [640 (12 + π) / (8 + π)] divided by x²

Remember that constant multipliers are just "kept aside" during the differentiation process.

Sorry, I’m still not sure how to get to the second equation.
If,
(8+π)x^2 + 640(12+ π)/x(8+ π)

= (8+ π)x^2 + 640(12+π) (1/x(8+π))

= (8+ π)x^2 + 640(12+π) (x(8+π))^-1

dS/dx
= (16+2π)x - 640(12+π) (x(8+π))^-2

= (16+2π)x - 640(12+π)/ (x(8+π))^2

= (16+2π)x - 640(12+π)/ (x^2)(8+π)^2

It’s different from your correct answer where yours is x^2(8+π) but mine is x^2(8+π)^2

Is there a working error?

Sorry, I’m still not sure how to get to the second equation.
If,
(8+π)x^2 + 640(12+ π)/x(8+ π)

= (8+ π)x^2 + 640(12+π) (1/x(8+π))

= (8+ π)x^2 + 640(12+π) (x(8+π))^-1

dS/dx
= (16+2π)x - 640(12+π) (x(8+π))^-2

= (16+2π)x - 640(12+π)/ (x(8+π))^2

= (16+2π)x - 640(12+π)/ (x^2)(8+π)^2

----------------------------------------------------

dS/dx
= (16+2π)x - 640(12+π) (x(8+π))^-2

This line is actually incorrect. The (8 + π) in the denominator (attached to the x) is a constant, and will therefore be taken out as a constant just like any other constant.

Alternatively, I note that the following is also possible:

S = ... + 640 (12 + π) / [x (8 + π)]
S = ... + constant TIMES 1 / [x (8 + π)]
S = ... + constant TIMES [x (8 + π)]^-1

dS/dx
= ... + constant TIMES (-1) [x (8 + π)]^-2 TIMES differentiate the chain [x (8 + π)]

{You probably missed out the chain rule on the [x (8 + π)]}

= --- + constant TIMES (-1) [x (8 + π)]^-2 TIMES d/dx [x (8 + π)]
= ... + constant TIMES (-1) [x (8 + π)]^-2 TIMES (8 + π)
= ... + constant TIMES (-1) x^-2 (8 + π)^-2 TIMES (8 + π)
= ... - constant TIMES (8 + π)^-1 TIMES x²

In your working, you missed out the chain rule to be applied when differentiating the expression

[x (8 + π)]^-1.

Chain rule will need to be applied in this case also, which will then cancel out the constant. This is why when I perform differentiation, I always consolidate the constant out (because I know it will remain unchanged later on).

Ahh I see thx!

What if the is a ‘x’ on the numerator?

For eg, d/dx ( 12x / (8+π)x )

Ahh I see thx!

What if the is a ‘x’ on the numerator?

For eg, d/dx ( 12x / (8+π)x )

------------------------------------------------------

Chain rule continues to apply as usual even as we differentiate expressions.

Alternatively, we can simplify the fractions to its simplest first before we extract any resulting constants.

In this case, d/dx {12x / [(8 + π) x]}
= d/dx [12 / (8 + π)] (because the x cancel out]
= 0 (because there is no x and we are simply differentiating a fixed number)

If the equation then is

dS/dx [ (12+x) / (8+π)x ]

Am I right to say

= - (12+x) / x^2(8+π)