What mathematical idea is the Leibniz notation for derivatives meant to convey?

Question

I know that mathematical notation is often designed to convey some of the mathematical ideas that it expresses -- but I am having trouble getting an intuition for the leibnitz notation like this

$\frac{d}{dx}h(x)$

I understand derivatives and limits. How is the Leibniz notation trying to reflect these concepts? Obviously we don't know exactly what Leibniz was thinking but I'm wondering if there is either a historical answer or a commonly-cited explanation?

Answer 1

Let $y(x)$ be a function.

Let $\Delta x$ be a number. Notice that $\Delta x=(x+\Delta x)-(x)$.

Define $\Delta y$ to be $y(x+\Delta x)-y(x)$.

$\dfrac{\operatorname d\!y}{\operatorname d\!x}$ is defined to be $\displaystyle\lim_{\Delta x\to0}\dfrac{\Delta y}{\Delta x}$. (Notice that, as $\Delta x$ goes to zero, $\Delta y$ goes to zero.)

Remember that Leibniz made his notation much earlier than the idea of a "limit". (The concept of a limit was due to Weierstraß.) He thought, not in terms of limits, but in terms of infinitesimals. To him, $\operatorname d\!x$ was an infinitesimal change in $x$ (much like how our $\Delta x$ is a real, not-infinitesimal change in $x$), and $\operatorname d\!y$ was the corresponding infinitesimal change in $y$ (like our $\Delta y$, but infinitesimal).

Leibniz said that infinitesimals like $\operatorname d\!x$ ("first order") were negligible compared to real numbers, infinitesimals like $\operatorname d\!x^2$ ("second order") were negligible compared to first order infinitesimals, etc. As you can tell, this theory isn't very rigorous, which is why Weierstraß had to put calculus on more rigorous footing later.

Answer 2

Leibniz notation denotes the idea of derivatives representing the gradient of a tangent curve.

The gradient of the secant line between $(x, h(x))$ and $(x + \delta x, h(x + \delta x))$ is given by $m = \frac{h(x + \delta x) - h(x)}{x + \delta x - x} = \frac{\delta h(x)}{\delta x}$, where $\delta h(x) = h(x + \delta x) - h(x)$. Then the gradient of the tangent to $h(x)$ at $x$ is given by $\frac{d h(x)}{dx} = \lim_{\delta x\rightarrow 0}\frac{\delta h(x)}{\delta x}$.

Then, once you notice that differentiation is essentially an operator from one function space to another, you can move the notation around a little to say that $\frac{d}{dx} h(x) = \frac{d h(x)}{dx}$, i.e. the operator $\frac{d}{dx}$ takes a function to its derivative with respect to $x$.