Solving Calculus: Derivative of x(t)

In summary, the conversation discusses the use of the chain rule in solving problems involving calculus and the concept of substitution. The speaker mentions that they are trying to repair their rusty calculus skills and have encountered some confusion with the use of du = dx*dt/dt. They also mention that the book's explanation seems absurd and instead suggests using dx = (dx/dt)*dt directly. The other speaker clarifies that it doesn't matter if x is a function of t and that the integral will still be in terms of x.
  • #1
EastWindBreaks
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Homework Statement


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Homework Equations

The Attempt at a Solution


I am trying to repair my rusty calculus. I don't see how du = dx*dt/dt, I know its chain rule, but I got (du/dx)*(dx/dt) instead of dxdt/dt, if I recall correctly, you cannot treat dt or dx as a variable, so they don't cancel out. so for (du/dx)(dx/dt) to become dxdt/dt, du/dx must equal to dt, which is not...
 

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  • #2
What the book has is a bit absurd. If ##u = x(t)##, then ##u## and ##x## are the same function. That's not really a substitution. Instead, you can use

##dx = \frac{dx}{dt} dt##

Directly.
 
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  • #3
PeroK said:
What the book has is a bit absurd. If ##u = x(t)##, then ##u## and ##x## are the same function. That's not really a substitution. Instead, you can use

##dx = \frac{dx}{dt} dt##

Directly.
thank you, I see, so it doesn't matter if x is a function of t, we just integrate it regularly?
 
  • #4
EastWindBreaks said:
thank you, I see, so it doesn't matter if x is a function of t, we just integrate it regularly?

If you have an integral in ##x##, you have an integral in ##x##. It doesn't matter that you can express ##x## as a function of another variable. In a way, you can always do that.
 
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1. What is the derivative of x(t)?

The derivative of x(t) is the instantaneous rate of change of the function at a specific time t. It represents the slope of the tangent line to the curve x(t) at that point.

2. Why is the derivative of x(t) important?

The derivative of x(t) is important because it helps us understand how a function changes over time. It is also used in many real-world applications, such as physics, engineering, and economics, to analyze and optimize various processes and systems.

3. How do you solve for the derivative of x(t)?

To solve for the derivative of x(t), 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 more complex functions by breaking them down into simpler parts.

4. What is the difference between the derivative of x(t) and the integral of x(t)?

The derivative of x(t) measures the rate of change of the function at a specific point, while the integral of x(t) measures the accumulation of the function over a given interval. In other words, the derivative tells us how the function is changing, while the integral tells us how much of the function exists within a certain range.

5. Can you find the derivative of x(t) at a specific point?

Yes, the derivative of x(t) can be evaluated at a specific point by plugging in the value of t into the derivative function. This will give you the slope of the tangent line to the curve x(t) at that particular point.

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