# Relevant equations the derivative equation

• myria36
In summary, the problem involves finding the rate of change of the surface area of a right circular cylinder with respect to its height, while the radius remains constant. The formula for surface area is given by A = 2πr(r + h), and the goal is to express the area in terms of h. The student attempted to distribute and manipulate the equation, but was unsure of how to take the derivative of h since it is only present in one term. The solution involves using the derivative of a constant function to simplify the equation and express the area in terms of h.
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myria36

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calculus - derivatives
1. 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.

2. Homework Equations
the derivative equation
dA / dr

3. The Attempt at a Solution
i first distributed the 2pir, to yield
2pir^2 + 2pirh
2pir^2(h/h) + 2pirh -- i added the (h/h), which is like multiplying by 1, to add h to the first term- I am not sure if this is correct, but i was just guessing.

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
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

Hint: What's the derivative of a constant function?

## What is the definition of the derivative equation?

The derivative equation is a mathematical formula that represents the rate of change or slope of a function at a specific point. It is written as f'(x) or dy/dx and is used to find the instantaneous rate of change of a function.

## What are the basic rules for finding the derivative of a function?

The basic rules for finding the derivative of a function are the power rule, product rule, quotient rule, and chain rule. The power rule states that the derivative of xn is nxn-1. The product rule states that the derivative of uv is u'v + uv'. The quotient rule states that the derivative of u/v is (u'v - uv')/v2. The chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x).

## How is the derivative equation used in real-world applications?

The derivative equation is used in many real-world applications, such as physics, economics, and engineering. It can be used to find the velocity of an object, the marginal cost and revenue in economics, and the rate of change of a chemical reaction in chemistry. It is also used to optimize functions and find maximum and minimum values.

## What is the difference between the derivative and the antiderivative?

The derivative and antiderivative are related, but they have different meanings. The derivative of a function represents the rate of change of that function, while the antiderivative represents the original function from which the derivative was taken. In other words, the derivative is the "slope" of a function, while the antiderivative is the "area" under the curve of that function.

## Can the derivative of a function be negative?

Yes, the derivative of a function can be negative. This means that the function is decreasing at that particular point. The derivative can also be positive, indicating that the function is increasing at that point. A derivative of zero means that the function is neither increasing nor decreasing at that point.

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