# Time Derivative: How Does 2x Differ from x² Differ?

• delve
In summary, the chain rule is used to take the two from the power of d/dt f(t) and take that as a coefficient, reduce the power by one, and then take the time derivative of d/dt f(t).
delve
I'm wondering, how does 2 multiplied by the first and second time derivatives of x equal the time derivative of the time derivative of x squared. Thanks.

Ummm... You mean,
$$2\frac{d^2}{dt^2} f(t) = \frac{d}{dt} \left(\frac{d}{dt} f(t) \right)^2$$
This would be done by the chain rule. That is,
$$\frac{d}{dx} (f(x))^2 = 2f(x)\frac{d}{dx}f(x)$$
In this sense, we take the two from the power of d/dt f(t) and take that as a coefficient, reduce the power by one, and then take the time derivative of d/dt f(t). I originally interpreted your question as
$$2\frac{d}{dt}\frac{d^2}{dt^2} f(t) = \frac{d}{dt} \left(\frac{d}{dt} f(t) \right)^2$$
but I do not feel this expression is true.

EDIT: Ok, let's fix this.

Ummm... You mean,
$$2\frac{d}{dt}f(t)\frac{d^2}{dt^2} f(t) = \frac{d}{dt} \left(\frac{d}{dt} f(t) \right)^2$$
This would be done by the chain rule. That is,
$$\frac{d}{dx} (f(x))^2 = 2f(x)\frac{d}{dx}f(x)$$
In this sense, we take the two from the power of d/dt f(t) and take that as a coefficient, reduce the power by one, and then take the time derivative of d/dt f(t).

Last edited:
Let v = dx/dt and a = dv/dt. Then I believe he means, why does 2va = d/dt (v^2)?

It immediately follows from the chain rule:

d/dt(v^2) = 2v*d/dt(v) = 2va

nicksauce said:
Let v = dx/dt and a = dv/dt. Then I believe he means, why does 2va = d/dt (v^2)?

It immediately follows from the chain rule:

d/dt(v^2) = 2v*d/dt(v) = 2va

Ahh gotcha, I missed an f in there when I wrote it out on paper.

$$2\frac{d}{dt}f(t)\frac{d^2}{dt^2} f(t) = \frac{d}{dt} \left(\frac{d}{dt} f(t) \right)^2$$

Of course that would make my original statement wrong too...

I need more coffee.

Awesome, thanks a lot for the help. I appreciate it.

## 1. What is the difference between 2x and x²?

The main difference between 2x and x² lies in their mathematical operations. 2x represents two times the value of x, while x² represents the square of x. This means that when x is multiplied by itself, the result is x². For example, 2x when x=3, would equal 6, while x² when x=3, would equal 9.

## 2. How do the graphs of 2x and x² differ?

The graphs of 2x and x² differ in shape and steepness. The graph of 2x is a straight line with a slope of 2, while the graph of x² is a parabola with a slope that continuously increases as x increases.

## 3. What are the domains and ranges of 2x and x²?

The domain for both 2x and x² is all real numbers, as they can take on any value for x. However, the range is different. The range of 2x is also all real numbers, while the range of x² is only positive real numbers (including zero).

## 4. How does the time derivative of 2x differ from the time derivative of x²?

The time derivative of 2x is simply 2, as the derivative of a constant times a variable is just the constant. On the other hand, the time derivative of x² is 2x, as it follows the power rule for derivatives. This means that the derivative of x^n is n*x^(n-1), in this case n=2.

## 5. In what real-world situations can 2x and x² be used to model relationships?

2x can be used to model relationships involving doubling or increasing by a constant factor. For example, if a company's profits double each year, 2x can be used to represent the annual profits. x² can be used to model relationships involving growth or acceleration, such as the distance of a falling object over time due to gravity.

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