When we write \[\frac{dy}{dx}\], we can think of it as a fraction-like object that tells us how \(y\) changes with respect to \(x\). This intuition is surprisingly powerful!

Consider the chain rule: \[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]

Notice how the \(du\) terms “cancel out” just like they would in a fraction. While this isn’t mathematically rigorous, it helps remember the rule.

Similarly, when we write \[\int f(x)dx\], we can think of \(dx\) as an infinitesimally small piece of \(x\) that we’re summing over. This helps explain why:

\[\int \frac{1}{x}dx = \ln x + C\]
Because when we differentiate \(\ln x \), the chain rule gives us \[\frac{1}{x} \cdot \frac{dx}{dx} = \frac{1}{x}\]

Even the fundamental theorem of calculus makes intuitive sense when we think about \(dx\) as tiny pieces:

\[\int_a^b f’(x)dx = f(b) - f(a)\]

We’re adding up all the tiny changes (\(f’(x)dx\)) to get the total change from \(a\) to \(b\).