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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:

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

- What is the correct integral of $\frac{1}{x}$?
- Hydrostatic pressure on a square
- What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\it J}_{0}\left(n\right)} ^2}{{n}^4}}$?
- Limit involving $(\sin x) /x -\cos x $ and $(e^{2x}-1)/(2x)$, without l'Hôpital
- Striking applications of integration by parts
- Proof of a Limit of a Function, given the Limits of the Multiplicative Inverses of the Function

also phrased as

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

e.g:

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

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

but

$(\text{derivative with respect to} (\lambda x \to 2))$ 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?

- Derivative of $f(x)$ with respect to a constant such as $1,2,3,\ldots 9999$
- 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)

- convert \ to lambdas
- convert d/dx to symbolic d/dx notation (not the worded derivative ones)
- convert arrows to arrows used in set theory/category theory
- 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).

- Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$?
- Find the indefinite integral $\int {dx \over {(1+x^2) \sqrt{1-x^2}}} $
- Interesting calculus problems of medium difficulty?
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- Calculate $\lim_{n\to\infty}(\sqrt{n^2+n}-n)$.
- Choosing a continuous function satisfying the mean value theorem
- Closed form for $\prod_{k=0}^n\binom{n}{k}x^ky^{n-k}$.
- Under what condition we can interchange order of a limit and a summation?
- A question about an exponential decay
- A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$

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.

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