# Derivatives and Partial Derivatives

• Trenthan
In summary, Trent believes that q1 and q2 are both functions of time, and that q1 and q2 are also related to each other through the third term in the Lagrange's equation.
Trenthan
ey guys

Generally i just do these without thinking, however i was checking some work today with a friend and he is adament i did my derivative wrong...

If i can double check with you

Well firstly
'c' is simply a constant
q1 and q2 are generalised coordinates
IZG1 is simply the moment of inertia
'T' being Kinetic energy

For those who don't notice what this is. Its apart of Lagrange's equation during the formation of the Equation of Motion

EDIT**, i should mention that q1 and q2 are both functions of time

$$T = \frac{1}{2}m_{1}c^{2}\dot{q_{1}}^2 + \frac{1}{2}I_{ZG1}\dot{q_{1}}^{2} + \frac{1}{2}m_{2}(\dot{q_{2}}^{2}+\dot{q_{1}}^{2}q_{2}^{2})+\frac{1}{2}I_{ZG2}\dot{q_{1}}^{2}$$

$$\\ \frac{\partial T}{\partial \dot{q_{1}}} = m_{1}c^{2}\dot{q_{1}} + I_{ZG1}\dot{q_{1}}+ m_{2}q_{2}^{2}\dot{q_{1}}+ I_{ZG2}\dot{q_{1}}$$

His version
$$\frac{d}{dt}\left ( \frac{\partial T}{\partial \dot{q_{1}}} \right ) = m_{1}c^{2}\ddot{q_{1}}+I_{ZG1}\ddot{q_{1}} + m_{2}\ddot{q_{1}}q_{2}^{2} + 2m_{2}q_{2}\dot{q_{1}} + I_{ZG2}\ddot{q_{1}}$$

Mine(below this), so unless I am mistaken it should have
$$\dot{q_{2}}$$
in the third term?

$$\frac{d}{dt}\left ( \frac{\partial T}{\partial \dot{q_{1}}} \right ) = m_{1}c^{2}\ddot{q_{1}}+I_{ZG1}\ddot{q_{1}} + m_{2}\ddot{q_{1}}q_{2}^{2} + 2m_{2}q_{2}\dot{q_{2}}\dot{q_{1}} + I_{ZG2}\ddot{q_{1}}$$

Now if someone can clarify which version is correct big help. I've gone back to my textbook however i don't have anything similar. Everything is linear so no 'squared' stuff

Cheers Trent, and thanks in advance

Last edited:
Your version is the correct one.

wohhh sorry i didnt make it clear

His version is the top

Mine is the lower one****

So you reffering to the top or bot

Soz lolz

Bottom. I guess you could have guessed this answer!

i agree with..the bottom equation is correct..

## 1. What are derivatives and partial derivatives?

Derivatives and partial derivatives are mathematical concepts used to measure the rate of change of a function with respect to its independent variables. Derivatives are used to calculate instantaneous rates of change, while partial derivatives are used to calculate rates of change in multi-variable functions.

## 2. What is the purpose of using derivatives and partial derivatives?

Derivatives and partial derivatives are used in various fields of science, such as physics, economics, and engineering, to model and analyze real-world phenomena. They can also be used to optimize functions and solve optimization problems.

## 3. How are derivatives and partial derivatives calculated?

The derivative of a function is calculated by taking the limit of the change in the function over the change in the independent variable, as the change in the independent variable approaches zero. Partial derivatives are calculated similarly, by taking the limit of the change in the function with respect to one variable, while holding all other variables constant.

## 4. What is the difference between a derivative and a partial derivative?

The main difference between a derivative and a partial derivative is that derivatives measure the rate of change of a function with respect to one independent variable, while partial derivatives measure the rate of change with respect to multiple independent variables, holding all other variables constant.

## 5. In what fields are derivatives and partial derivatives commonly used?

Derivatives and partial derivatives are commonly used in fields such as physics, economics, engineering, statistics, and finance. They are also utilized in machine learning and data analytics to optimize models and make predictions.

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