# 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

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

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?

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).
• So it always has to be a free variable, and the function MUST be expressed through the free variable (which can be a function which must also be free)? (Response to apparently deleted post). – Dmitry Apr 13 '13 at 22:51
• I've edited your question to use $\LaTeX$ (the math formatting we use). Please make sure the post still represents your original intent. A great resource for learning formatting here is this meta question. – apnorton Apr 13 '13 at 23:18
• We do not differentiate expressions. We differentiate functions. The notation $d/dx (\mbox{expression in }x)$ simply means "define a function using the given expression, differentiate that function, and evaluate at $x$". As for differentiating "with respect to" something other than a free variable - I've never seen anyone try to define or use this. – wj32 Apr 13 '13 at 23:37
• I think that notion is confusing because the result of the d/dx is intended to be the function of the slope at any given point of the given function, howerver (d/dx)x is a constant(1 in this case), which is not a function, and performing function application of a constant on anything is illogical(eg, lisp would yell at you for trying to do (1 2)), so I am trying to distance myself from this definition. The alternative defines (d/dx)f = (\x -> (x -> f'x)) which when applied onto x, provides a function (\x -> (x -> 1)), which causes no contradictions. – Dmitry Apr 13 '13 at 23:43

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.
• Derivative of constant w.r.t. constant would lead to $\lim_{h\to0}(0/0)$, which is undefined. Derivative of function w.r.t. constant, as shown above, does not exist. I don't see how this can be interpreted as a finite difference. And I don't know what to make of "does that mean that $h\to0$ is essentially synonymous to $dx$". – Gerry Myerson Apr 14 '13 at 0:10
• They make sense, if you define them in a sensible way. You could, for example, define $df(x)$ to mean $f'(x)\,dx$ and, under certain circumstances, you might find it useful to do so. If I were teaching an intro Calculus course, though, I'd probably steer clear of such notations --- students have enough trouble with the usual ones, they don't need more. – Gerry Myerson Apr 14 '13 at 6:27