Partial Derivatives: Solving T(x,t)=S(n) Chain Rule Mistake

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In summary, 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 the other variables constant. To solve for a partial derivative, the function is differentiated with respect to the specific variable of interest, using basic rules of differentiation such as the power rule and chain rule. The chain rule is a crucial tool in finding the derivative of a composite function, and is often used in calculating partial derivatives of functions with variables that are themselves functions. Mistakes in applying the chain rule can lead to incorrect results, making careful application and double-checking important. Partial derivatives have a wide range of applications in fields such as physics, engineering, and economics, where
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astoria
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T(x,t)=S(n) where n is some given function of x and t.


Why is (partial dT by partial dt)=dS/dn*(partial dn by partial dt)

What happens to the extra (all partials) (dS/dx)*(dx/dt)

I guess I'm misunderstanding the chain rule in partial derivatives but can someone point out my mistake.

Thanks
 
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  • #2
S is a function of only n, so you don't differentiate it with respect to x. Where ∂x/∂t will show up is when you differentiate n with respect to t because n=n(x,t).
 

1. What is a partial derivative?

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 the other variables constant. It is denoted by ∂ (pronounced "partial") and is often used in fields such as physics, engineering, and economics to analyze how a system responds to changes in multiple variables.

2. How do you solve for partial derivatives?

To solve for a partial derivative, you need to differentiate the function with respect to the specific variable you are interested in, treating all other variables as constants. This involves using the basic rules of differentiation, such as the power rule and chain rule. The resulting partial derivative will be a new function that describes the rate of change of the original function with respect to the chosen variable.

3. What is the chain rule and how is it used in partial derivatives?

The chain rule is a rule in calculus that allows us to take the derivative of a composite function. In the context of partial derivatives, it is used to calculate the derivative of a function with respect to a variable that is itself a function of another variable. In other words, it helps us to find the rate of change of a function with respect to a variable that is changing due to the change of another variable.

4. What is the importance of avoiding chain rule mistakes in partial derivatives?

The chain rule is a crucial tool in solving for partial derivatives, and making a mistake in its application can lead to incorrect results. In some cases, even a small error in the chain rule can result in a vastly different solution. Therefore, it is important to carefully apply the chain rule and double-check for any mistakes to ensure accurate results.

5. How are partial derivatives used in real-world applications?

Partial derivatives have a wide range of applications in various fields, including physics, engineering, economics, and statistics. They are used to analyze the behavior of complex systems and to make predictions about how these systems will respond to changes in different variables. For example, they are used in economics to analyze the relationship between factors such as price, demand, and supply, and in physics to understand the motion and behavior of objects in three-dimensional space.

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