madah12 said:cant you just say [tex] \frac{\partial T}{\partial x} = \frac{\partial T}{\partial r} * \frac{\partial r}{\partial x} [/tex]
but since T is function of r you can write [tex] \frac{\partial T}{\partial r} [/tex] as dT/dr I am not sure this is correct though.
A partial derivative is a mathematical concept used in multivariable calculus to measure the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is represented by the symbol ∂ (pronounced "partial"), and is used to calculate how a small change in one variable affects the value of the function.
To calculate a partial derivative, you use the same rules as for regular derivatives. You start by taking the derivative of the function with respect to the variable you are interested in, treating all other variables as constants. This means you can use the power rule, product rule, quotient rule, and chain rule as needed. Once you have the derivative, you can plug in the values for the variables to get the specific partial derivative at that point.
The main difference between a partial derivative and a regular derivative is that a partial derivative calculates the change in a function with respect to one variable, while holding all other variables constant. A regular derivative, on the other hand, calculates the change in a function with respect to one variable while allowing all other variables to vary. Another difference is that a partial derivative can result in a function of multiple variables, while a regular derivative results in a single value.
Partial derivatives are used in various fields of science and engineering to analyze how a function changes in response to changes in its variables. They are particularly useful in understanding systems with multiple variables, such as in thermodynamics, economics, and physics. They also play a critical role in optimization problems, where the goal is to find the maximum or minimum value of a function.
Partial derivatives have many real-life applications. For example, in economics, they are used to analyze the effects of changing variables on supply and demand. In physics, they are used to understand how physical quantities such as velocity and acceleration change with respect to time. In engineering, they are used to optimize designs and understand the relationships between various parameters. In machine learning, they are used to optimize algorithms and train models. Overall, partial derivatives are an essential tool for understanding and modeling complex systems in various scientific and engineering fields.