Calculus Solver What dx and dy/dx mean
dy/dx is the derivative of y with respect to x: the rate at which y changes as x changes. It is Leibniz notation, and it is read as one symbol rather than a fraction of two numbers. On its own, dx marks which variable you are differentiating or integrating with respect to.
Reading dy/dx
dy/dx means the derivative of y with respect to x. The d is not a variable being multiplied: dy/dx is a single symbol standing for a limit, the limit of Δy/Δx as Δx shrinks to zero.
That is why you cannot cancel the d's, and why dy/dx is safest read aloud as "the derivative of y with respect to x" rather than "dy over dx".
What dx means by itself
In an integral, dx says which variable you are integrating with respect to, and where the integrand ends. In ∫2x dx the dx tells you x is the variable, so any other letter present is a constant along for the ride.
Informally dx is an infinitesimally small step in x. That reading is what makes the notation so useful: ∫f(x) dx looks like adding up strips of height f(x) and width dx, which is exactly the picture behind the integral.
dy/dx against dx/dy
They are different derivatives, taken in opposite directions. dy/dx measures how y responds to a change in x; dx/dy measures how x responds to a change in y.
For well behaved functions they are reciprocals: dx/dy = 1/(dy/dx). If y = x³ then dy/dx = 3x², so dx/dy = 1/(3x²). This is one of the few places where treating the notation like a fraction gives the right answer, which is exactly why Leibniz chose it.
The same idea in different notations
| Leibniz | Lagrange | Means |
|---|---|---|
| dy/dx | f'(x) | First derivative |
| d²y/dx² | f''(x) | Second derivative |
| dy/dt | ẏ | Derivative with respect to time |
| ∂z/∂x | — | Partial derivative, other variables held fixed |
Questions
Is dy/dx a fraction?
Not literally: it is the limit of a ratio, written as one symbol. It behaves like a fraction often enough that the notation was worth keeping, which is why the chain rule and separation of variables look like cancelling.
What does the dx at the end of an integral do?
It names the variable of integration and marks where the integrand stops. Without it, ∫2xy would be ambiguous about whether x or y is the variable.
What is the difference between dy/dx and Δy/Δx?
Δy/Δx is the slope between two separate points, an average rate of change. dy/dx is what that ratio approaches as the two points come together: the slope at a single point.
More calculus reference
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