Think of differentials of picking apart the “fraction” dydx we learned to use when differentiating a function.
We learned that the derivative or rate of change of a function could be written as dydx=f′(x), where dy is an infinitely small change in y, and dx (or Δx) is an infinitely small change in x. It turns out that if f(x) is a function that is differentiable on an open interval containing x, and the differential of x (dx) is a non-zero real number, then dy=f′(x)dx (see how we just multiplied both sides by dx)? And I won’t get into this at this point, but the differential of y can be used to approximate the change in y, so Δy≈dy.
We can use differentials to perform linear approximations of functions (we did this here with tangent line approximation) with this formula that looks similar to a point-slope formula (remember that the derivative is a slope): y−y0=f′(x0)(x−x0), or f(x)−f(x0)=f′(x0)(x−x0), which means f(x)=f(x0)+f′(x0)(x−x0). And remember that the variables with subscript “0” are the “old” values. Think of the equation as the “new y” equals the “old y” plus the derivative at the “old x” times the difference between the “new x” and the “old x”.
We can also use differentials in Physics to estimate errors, say in physical measuring devices. In these problems, we’ll typically take a derivative, and use the “dx” or “dy” part of the derivative as the error. Then, to get per cent error, we’ll divide the error by the total amount and multiply by 100.