Simple question about derivatives

In summary, the conversation discusses the concept of taking the derivative of a function with respect to a constant and whether it is the same as changing the argument of the function. The individual asking the question provides examples and asks for confirmation, while the other person clarifies the concept and expresses confusion about taking the derivative with respect to a constant. The conversation then delves into a specific example using Hamilton's equations and discusses the Jacobian for this transformation.
  • #1
Pacopag
197
4

Homework Statement


Suppose you have a function [tex]f(x)[/tex].
You take the derivative with respect to x and evaluate it at some point [tex]x_0[/tex].
i.e. [tex]{{df(x_{0})}\over{dx}}[/tex] (e.g. first coeff in taylor series)
Is this the same as changing the argument of f to x_0, i.e. writing [tex]f(x_0)[/tex],
then just taking the derivative with respect to x_0?
In fewer words, can I write
[tex]{{df(x_{0})}\over{dx}}={{df(x_{0})}\over{dx_0}}[/tex].


Homework Equations





The Attempt at a Solution


If I just consider a bunch of examples, it always seems to be true. It seems like a trivial fact, but I would just like someone to confirm this for me.

Thanks.
 
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  • #2
No,

[tex]\frac{df(x_0)}{dx}[/tex] may be read as the derivative of f(x) with respect to x, evaluated at [tex]x = x_0[/tex]. I am confused as to how you were taking the derivative of something with respect to a constant.
 
  • #3
You're right. It seems highly unorthodox.
Ok. Consider the taylor series (keeping only first order)
[tex]q(\Delta t)=q(0)+{{\partial H(q,p)}\over{\partial p}}\Delta t[/tex]
[tex]p(\Delta t)=p(0)-{{\partial H(q,p)}\over{\partial q}}\Delta t[/tex]
where the derivatives are evaluated at t=0. I used hamilton's equation.
[tex]{{dq}\over{dt}}={{\partial H}\over{\partial p}}[/tex]
[tex]{{dp}\over{dt}}=-{{\partial H}\over{\partial q}}[/tex]
Now regard this as a transformation from the old coords q(0), p(0) to the new coords q(Delta t), p(Delta t).
What is the Jacobian for this tranformation?
 

What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It measures how much a function changes with respect to its input.

Why are derivatives important?

Derivatives are important because they allow us to find the slope of a curve at a specific point, which is useful in many real-world applications such as physics, economics, and engineering. They also help us to optimize functions and solve problems involving rates of change.

How do you find a derivative?

To find a derivative, you can use the formula for the derivative, which is the limit of the difference quotient as the change in input approaches zero. Alternatively, you can use differentiation rules and techniques such as the power rule, product rule, and chain rule.

What is the difference between a derivative and an integral?

A derivative measures the rate of change of a function, while an integral measures the accumulation of a function over an interval. In other words, a derivative tells us how fast the function is changing, and an integral tells us how much the function has changed over a given period.

What are some real-life applications of derivatives?

Derivatives have many real-life applications, including calculating velocity and acceleration in physics, finding maximum and minimum values in optimization problems, and determining the marginal cost and revenue in economics. They are also used in fields such as engineering, finance, and computer science.

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