Derivative transform
Convert a time derivative into algebra while preserving an initial value.
Derivative transform is one of 1 laplace transforms formula in the differential equations section of this library, and it is used at university level.
Why derivative transform works
Apply integration by parts to the transform of f'(t). The boundary piece contributes −f(0) and the remaining integral is s times the transform of f itself. So differentiating in time becomes multiplying by s in the transform world, and the starting value gets carried along instead of being lost.
What each symbol means
$F(s)=\mathcal L\{f(t)\}$.
Derivative transform: when it holds
$f$ must satisfy conditions ensuring its Laplace transform exists.
When it stops applying
It needs a function whose transform actually exists, meaning f grows no faster than some exponential. A function like e^(t²) outruns every e^(−st) and has no transform, so the rule has nothing to work with. A jump at t = 0 also forces care over whether f(0) means the value just before or just after.
Derivative transform: a worked example
This turns many initial-value differential equations into equations for $F(s)$.
The mistake to avoid
What people do: Students drop the −f(0) term and write the transform of the derivative as just sF(s).
Why it goes wrong: That term is where the initial condition lives. Without it the transform describes a problem that always starts at zero, so any nonzero starting value silently disappears.
Do this instead: Keep the initial value attached. Check with f(t) = e^(2t), whose transform is 1/(s − 2): the rule gives s/(s − 2) − 1 = 2/(s − 2), which matches transforming f'(t) = 2e^(2t) directly.
Derivative transform: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $F(s)=\mathcal L\{f(t)\}$.
- Check the conditions before substituting. $f$ must satisfy conditions ensuring its Laplace transform exists.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most differential equations slips.
Where this formula fits
- Subject
- Differential Equations formulas — 8 entries in this library
- Topic
- Laplace transforms
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where derivative transform comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Laplace Transform Calculator — check your substitution and the value it produces.
- Study differential equations — the subject guide that explains the ideas these formulas compress.
- Differential Equations Practice — questions that make you retrieve the formula instead of recognising it.
- All 8 differential equations formulas — the full grouped reference, or the complete formula library.
Questions about derivative transform
What is the rule for the second derivative?
Apply the same rule twice: the transform of f'' is s²F(s) − s f(0) − f'(0). That is why a second-order initial-value problem needs both the starting position and the starting velocity.
Why bother converting to a transform at all?
Because it turns calculus into algebra. A differential equation becomes an ordinary equation in F(s) that you solve with the usual rearranging, then convert back at the end.
What if my initial condition is given at t = 1?
Shift the time variable so the condition sits at zero. Define a new function g(t) = f(t + 1), solve for g, then shift the answer back at the end.
How do I get back to a function of t?
By inverting the transform, usually with partial fractions and a table. That last step is often the hardest part of the whole method, since the algebra can be heavy.