Question

The total surface area of a right circular cylinder of radius r and height h is
A = 2πrh + 2πr^2. Find dh/dr if A has a fixed value. Answer (−2πr^2−A)/2πr^2

Answer 1

A = 2πrh + 2πr^2
dA/dr = 2π(h + r dh/dr + 2r)
2π(h + r dh/dr + 2r) = 0
r dh/dr = -(h+2r)
dh/dr = -h/r - 2

See whether you can massage that into the desired form.

Answer 2

A = 2 π r h + 2 π r²

A - 2 π r² = 2 π r h

2 π r h = A - 2 π r²

h = ( A - 2 π r² ) / 2 π r =

A / 2 π r - 2 π r² / 2 π r =

A / 2 π r - ( 2 π / 2 π ) r² / r =

A / 2 π r - 1 ∙ r =

A / 2 π r - r =

( A / 2 π ) ∙ 1 / r - r =

( A / 2 π ) ∙ r ⁻¹ - r


h = ( A / 2 π ) ∙ r ⁻¹ - r


dh / dr = ( A / 2 π ) ∙ d ( r ⁻¹ ) / dr - dr / dr =

( A / 2 π ) ∙ ( - 1 ) ( r ⁻¹⁻¹ ) - 1 =

( A / 2 π ) ∙ ( - 1 ) ( r ⁻² ) - 1 =

- ( A / 2 π ) ∙ ( r ⁻² ) - 1 =

- ( A / 2 π ) ∙ 1 / r ² - 1 =

- ( A / 2 π ) / r ² - 1 =

- A / 2 π r ² - 1


dh / dr = - A / 2 π r ² - 1


If you really must write ( - 2 π r ² - A ) / 2 π r ²

then

dh / dr = - A / 2 π r ² - 1 =

- A / 2 π r ² - 2 π r ² / 2 π r ² =

( - A - 2 π r ² ) / 2 π r ² =

( - 2 π r ² - A ) / 2 π r ²


But expression

dh / dr = - A / 2 π r ² - 1

is simpler