Properties of differentials

The differential of a function $g(t)$ is written: $dg = g'(t) dt$.
Since this formula just uses the derivative of $g$, differentials obey all the usual rules of derivatives.
In the following $f$ and $g$ are differentiable functions of $t$ and $c$ and $n$ are real constants.

• If $g(t) = c$ is a constant function, then $dg = 0$, since $g'(t) = 0$.
• $d(f + g) = df + dg$:
  $$d(f + g) = (f + g)'dt = (f' + g')dt = f'dt + g'dt = df + dg.$$

• $d(f - g) = df - dg$:
  $$d(f - g) = (f - g)'dt = (f' - g')dt = f'dt - g'dt = df - dg.$$

• $d(fg) = fdg + gdf$:
  $$d(fg) = (fg)' dt = (f'g + fg')dt = f(g'dt) + g(f'dt) = fdg + g df.$$

• $d(\frac{1}{g}) = - \frac{dg}{g^2}$, provided $g\ne 0$:
  $$d(\frac{1}{g}) = (g^{-1})'dt = - g^{-2}g' dt = - g^{-2} dg = - \frac{dg}{g^2}.$$

• $d(\frac{f}{g}) = \frac{gdf - fdg}{g^2}$, provided $g\ne 0$:
  $$d(\frac{f}{g}) = (\frac{f}{g})'dt = \frac{gf' - fg'}{g^2} dt = \frac{gf'dt - fg'dt}{g^2} = \frac{gdf - fdg}{g^2}.$$

• Some differentials of standard functions:
  • $dt^n = nt^{n-1}dt$
  • $d(\sin(t)) = \cos(t) dt$,
  • $d(\cos(t)) = - \sin(t) dt$,
  • $d(\tan(t)) = \sec^2(t) dt$,
  • $d(e^t) = e^t dt$,
  • $d\ln(t) = \frac{dt}{t}$.
• $d (f^n) = nf^{n-1} df$, provided $f^n$ is defined.
  This is the differential version of the generalized power rule.
  Examples:
  $$d((1 + t^3)^3 = 3(1 + t^2)^2 3t^2 dt,$$
  $$d((\sin(t))^3) = 3(\sin(t))^2 \cos(t) dt.$$
  $$d(\sqrt{1 - t^2}) = \frac{1}{2}(1 - t^2)^{-\frac{1}{2}} (-2t) dt = - \frac{tdt}{\sqrt{1 - t^2}}.$$

• $d(f\circ g) = (f'\circ g) dg$.
  This is the differential version of the chain rule.
  We can also write this formula as:
  $$d(f(g(t)) = f'(g(t)) dg = f'(g(t)) g'(t) dt$$
  Examples:
  $$d(\sin(1 + t^3)) = \cos(1 + t^3)d(1 + t^3) = \cos(1 + t^3)3t^2 dt,$$
  $$d(\ln(\cos(t))) = \frac{1}{\cos(t)}d(\cos(t) = \frac{1}{\cos(t)} (- \sin(t) dt) = - \tan(t) dt,$$
  $$d(e^{4t^2}) = e^{4t^2}d(4t^2) = e^{4t^2}8t dt.$$