Finding Derivatives with a Constant Radius

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SUMMARY

The discussion focuses on finding the derivative of the surface area of a right circular cylinder, represented by the formula A = 2πr(r + h), with respect to height (h) while keeping the radius (r) constant. The user attempts to differentiate the equation but struggles with the presence of h in only one term. The correct approach involves applying the derivative rules appropriately, leading to the result dA/dh = 2πr, as the terms involving h simplify correctly when differentiated.

PREREQUISITES
  • Understanding of basic calculus, specifically differentiation.
  • Familiarity with the formula for the surface area of a cylinder.
  • Knowledge of the product rule in differentiation.
  • Ability to manipulate algebraic expressions involving variables.
NEXT STEPS
  • Study the product rule in calculus for differentiating products of functions.
  • Practice finding derivatives of multi-variable functions.
  • Explore the implications of keeping variables constant during differentiation.
  • Review examples of surface area calculations for different geometric shapes.
USEFUL FOR

Students studying calculus, particularly those focusing on derivatives and applications in geometry, as well as educators looking for examples of derivative problems involving geometric formulas.

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Homework Statement



the total surface area of a right circular cylinder is given by the formula: (A = 2Pir(r + h) ).
where r is the radius and h is the height.
sub part a) find the rate of change of A with respect to h is r remains constant

i know how to take derivatives. the only thing is that in this case, I am not sure how to take the derivative of h since it is only present in one term.

Homework Equations


the derivative equation


The Attempt at a Solution


i first ditributed the 2pir, to yield
2pir^2 + 2pirh
2pir^2(h/h) + 2pirh
h (2pir^2 h^-1 + 2pi r)
now i am stuck here. i can't take the derivative of all the h's in my problem, because one h is still present in the equation.
**below is my attempt to still work with it.
dA/dh = 1 times [-1(2pir^2h^-2) + 2pir
final answer: (-2pir^2h^-2) + 2pir
please can someone guide me on the technique i should use for getting the area to be in terms of h. any and all replies are welcome and appreciated
 
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