derivative with respect to constant.

I have been beating my head against this question for quite some time, I do not know whether it has been asked before, but I can’t find any information about it!

I am taking Calculus 1 course and I cannot grasp the concept of a derivative. From what I understand, a derivative is a function with the following signature:

$$(\text{derivative with respect to particular free variable}) :: (\lambda x \to (f)\; x) \to (\lambda x \to (f’) x)$$

also phrased as

$$(\text{derivative with respect to particular free variable}) = ((\lambda x \to (f) x) \to (\lambda x \to (f’) x))$$


$$(\text{derivative with respect to x}) (\lambda x \to x^2) = (\lambda x\to 2\cdot x)$$

This makes sense but one thing bothers me: what does “derivative with respect to x” mean?
In particular, in single variable Calculus this notation assumes $x$ is always a particular variable such as ['a'..'z'].

This works fine for basic derivatives such as:
$$(\text{derivative with respect to x}) (\lambda x \to \ln x) = (\lambda x \to \tfrac{1}{x})$$

What I would like to understand is: Why does (derivative with respect to x) make sense
$(\text{derivative with respect to} (\lambda x \to 2))$ and

$(\text{derivative with respect to} (\lambda x \to \ln(x)) $
do not seem to make any sense to me.

in classical terms, I cannot do (derivative of $\ln x$ with respect to $1$) nor (derivative of $\ln x$ with respect to $\ln x$) without my head starting to hurt, because those concepts were not taught to me yet, or I have not payed enough attention to understand them.

Can somebody please explain what the following two really mean?

  1. Derivative of $f(x)$ with respect to a constant such as $1,2,3,\ldots 9999$
  2. Derivative of $f(x)$ with respect to a function such as $\ln(x)$, $\sin(x)$, $\cos(x)$

Thanks ahead of time, this has been bothering me for quite a few years!

$\langle$Editor’s note: I’ve left the following in the post for archival’s sake.$\rangle$

PS: I am terrible at formatting so to the great ones responsible for
formatting noob’s questions (I thank you much for your work)

  1. convert \ to lambdas
  2. convert d/dx to symbolic d/dx notation (not the worded derivative ones)
  3. convert arrows to arrows used in set theory/category theory
  4. keep the “(derivative of … with respect to …)” as they are, as I have no idea how to express them differently, dA/dB doesn’t seem to
    make sense to me since derivatives are taught to be polymorphic
    function rather than a function of two variables, and division only
    makes it even more confusing due to the abuse of notation. (Feel free
    to give me a link to study formatting, I can’t find it).

Solutions Collecting From Web of "derivative with respect to constant."

Derivatives are usually defined in terms of limits. The derivative of $f(x)$ with respect to $g(x)$ can be defined as $$\lim_{h\to0}{f(x+h)-f(x)\over g(x+h)-g(x)}$$ provided the limit exists. In the case $g(x)=x$, this reduces to the familiar formula for the derivative of $f(x)$ with respect to $x$, $$\lim_{h\to0}{f(x+h)-f(x)\over h}$$ In the case where $g(x)$ is a constant, the denominator $g(x+h)-g(x)$ is identically zero, so the limit n’existe pas. This could explain why no one ever differentiates with respect to a constant.

dy/dx in words means change in y with one unit small change in x so in cases where x is a constant that simply means there is no change in x against which you would otherwise look at change in y so simply tht implies dx=0 and hence dy/dx is not defined.