Derivative vs Differential

Derivative and differential are closely related but distinct mathematical objects, and confusing them is the source of many subtle calculus errors.

Derivative

The derivative $f'(x)$ (or $\frac{dy}{dx}$) is a function that gives the rate of change of $f$ at each $x$. For $f(x) = x^2$, $f'(x) = 2x$.

Numerically: at $x = 3$, $f'(3) = 6$ — the slope of the tangent line at that point.

Differential

The differential $dy$ is an infinitesimal change in $y$ corresponding to an infinitesimal change $dx$ in $x$:

$dy = f'(x) \, dx$

For $y = x^2$: $dy = 2x \, dx$.

Differentials let you write derivatives as ratios of infinitesimals — useful in substitution ($u$-substitution in integrals: $du = u'(x) dx$) and in separation of variables for differential equations.

When the difference matters

In integrals: $\int 2x \, dx$ uses the differential $dx$, not the derivative.

In implicit differentiation: from $x^2 + y^2 = 25$, take differentials: $2x \, dx + 2y \, dy = 0$, then solve for $\frac{dy}{dx}$.

In physics: $dW = F \, dx$ (work as differential), not "work equals derivative of force."

Linear approximation

$dy$ also serves as a linear approximation to $\Delta y$ (actual change) for small $dx$:

$\Delta y \approx dy = f'(x) \, dx$

This is the basis of error propagation, Newton's method, and the linear-approximation foundation of all of calculus.

Verdict

Use derivative $f'(x)$ when you want a rate / function. Use differential $dy = f'(x) dx$ when you want an infinitesimal change, especially in integrals, substitution, or DEs.