# Time derivatives where there's no explicit time dependence

• Pythagorean
In summary, to take the time derivative of a function sin(theta-phi) where both theta and phi are time-dependent, you can use the multivariate chain rule. This involves finding the partial derivatives of the function with respect to theta and phi, and multiplying them by their respective time derivatives. It is important to note that this method assumes theta and phi are functions of time only.

#### Pythagorean

Gold Member
Say I have a funciton sin(theta - phi)

theta and phi are both time-dependant. How do I take the time derivative of this function? Is there a general notation or do I have to assume theta = w1t and phi = w2t and go with that?

You use the multivariate chain rule:

$$\frac{d}{dt}f(\theta,\phi) = \frac{\partial f}{\partial \theta}\frac{d \theta}{dt} + \frac{\partial f}{\partial \phi}\frac{d \phi}{dt}$$

Of course, you do then need to know the time derivatives of theta and phi. (Note that this also assumes theta and phi are functions of time only).

ah, thank you much, that's exactly what I was looking for.

## 1. What are time derivatives?

Time derivatives are mathematical operations that measure the rate of change of a physical quantity with respect to time. They are commonly used in physics and other scientific fields to describe the behavior of systems over time.

## 2. What is explicit time dependence?

Explicit time dependence refers to a situation where a physical quantity explicitly depends on time in a mathematical expression. This means that the quantity's value changes with time and is directly affected by it.

## 3. Can time derivatives be calculated when there is no explicit time dependence?

Yes, time derivatives can still be calculated even when there is no explicit time dependence. In this case, the time derivative is obtained by taking the derivative of the quantity with respect to time, without considering any explicit time dependence in the mathematical expression.

## 4. How do time derivatives without explicit time dependence impact scientific research?

Time derivatives without explicit time dependence can have a significant impact on scientific research. They allow scientists to study and understand systems that do not have a direct dependence on time, which can lead to new discoveries and advancements in various fields.

## 5. What are some examples of systems that exhibit time derivatives without explicit time dependence?

One example is a simple pendulum, where the angle of the pendulum's swing does not explicitly depend on time, but its velocity and acceleration do. Another example is a chemical reaction, where the concentration of reactants and products may change over time, but there is no explicit time dependence in the reaction rate equation.