Derivatives Problem (Calculus I)

In summary, to find the derivative with respect to time for a rectangle, first assume that both the base and height of the rectangle are functions of time. Then, use the product rule to find the derivative of the area formula A=bh. The result should be dA/dt = (b(dA/dh)) + (h(dA/db)), where b and h are both functions of time.
  • #1
PotentialE
57
0

Homework Statement


find d/dt for a rectangle.

Homework Equations


A=bh
product rule for derivatives (the first times the derivative of the second plus the second times the derivative of the first)
Chain rule for derivatives

The Attempt at a Solution


If b is a constant, then I know that dA/dh = b (this was the previous problem in which I could solve)
my issue with the current problem is that this variable t that I'm supposed to take the derivative with respect to is not in the problem- so how would I go about doing this?

My attempt:
d/dt (A=bh)
dA/dt = bh
dA/dt = (b(dA/dh)) + (h(dA/db))
My only problem with this solution is that I think there should be a d(b or h)/dt on the right side of the equation because of the chain rule

could someone please explain where/why the variable t comes in and what the correct answer is? Thank you!
 
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  • #2
PotentialE said:
find d/dt for a rectangle.

I have no clue what this means. Can you quote the full problem as it appears in your book?
 
  • #3
that's my problem too... that IS the quote!

"Find d/dt for a rectangle" and then the formula given is A=bh
so find d/dt for A=bh
 
  • #4
Well, that makes no sense. I advice asking your teacher for more explanations.
 
  • #5
On a previous problem:
Find d/dt for a square

and the answer was:
d/dt (A=s2)

dA/dt = 2s (ds/dt)

perhaps you understand this one? (I don't)
well i know the derivative of s2 = 2s. so we've taken the derivative of A with respect to t... so perhaps multiplying 2s by ds/dt is part of the chain rule or something... I don't really know where ds/dt came from
 
  • #6
PotentialE said:

Homework Statement


find d/dt for a rectangle.


Homework Equations


A=bh
product rule for derivatives (the first times the derivative of the second plus the second times the derivative of the first)
Chain rule for derivatives

The Attempt at a Solution


If b is a constant, then I know that dA/dh = b (this was the previous problem in which I could solve)
my issue with the current problem is that this variable t that I'm supposed to take the derivative with respect to is not in the problem- so how would I go about doing this?

My attempt:
d/dt (A=bh)
dA/dt = bh
dA/dt = (b(dA/dh)) + (h(dA/db))
My only problem with this solution is that I think there should be a d(b or h)/dt on the right side of the equation because of the chain rule

could someone please explain where/why the variable t comes in and what the correct answer is? Thank you!
Assume that b and h are both functions of time.

There should be no derivatives of A on the right hand side of the following:

dA/dt = (b(dA/dh)) + (h(dA/db))

Use the product rule to find [itex]\displaystyle \frac{d}{dt}(b\,h)\ .[/itex
 

What are derivatives in calculus?

Derivatives in calculus are a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of a tangent line to the curve of a function at a given point.

What is the purpose of solving derivatives problems?

Solving derivatives problems helps us to understand the behavior of a function and its rate of change. It is also useful in finding the maximum and minimum values of a function, which are important in many real-world applications.

How do you find the derivative of a function?

To find the derivative of a function, you can use the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules allow you to find the derivative of a function by using algebraic manipulation of its equation.

What are some common applications of derivatives?

Derivatives have many applications in various fields, including physics, economics, and engineering. Some common applications include optimization problems, finding velocities and accelerations in kinematics, and determining marginal costs and revenue in economics.

What are some common mistakes to avoid when solving derivatives problems?

Some common mistakes to avoid when solving derivatives problems include forgetting to use the chain rule, incorrectly applying the product and quotient rules, and not simplifying the final answer. It is also important to carefully differentiate between variables and constants in the problem.

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