Summary
- The derivatives of sinh x and cosh x fall straight out of their exponential definitions.
- All six hyperbolic derivatives mirror the trig ones — except cosh x and sech x, where the sign is opposite to what you'd expect from trig.
- Chain rule, product rule and quotient rule all work exactly as normal on hyperbolic functions.
- Inverse function derivatives (trig hyperbolic) are all found the same way: set y = f⁻¹(x), rewrite as f(y) = x, differentiate implicitly, substitute back using a Pythagorean-type identity.
- This gives you a table of six "new" standard derivatives worth memorising outright.
1. Derivatives of hyperbolic functions
Recall from Chapter 1 that hyperbolic functions are built directly from
e^x:
Because these are just combinations of exponentials, and we already know
d/dx(eˣ) = eˣ and d/dx(e⁻ˣ) = −e⁻ˣ,
differentiating sinh and cosh
is almost embarrassingly easy — and this is exactly why they swap into each other
instead of picking up a minus sign like sin and cos do.
Derivation
→ Start with y = sinh x = ½(eˣ − e⁻ˣ)
→ Differentiate term by term: dy/dx = ½(eˣ − (−e⁻ˣ)) = ½(eˣ + e⁻ˣ)
→ That's exactly the definition of cosh x.
→ d/dx(sinh x) = cosh x
→ Do the same for cosh x = ½(eˣ + e⁻ˣ): differentiating gives ½(eˣ − e⁻ˣ)
→ d/dx(cosh x) = sinh x
Now for tanh x, we use the quotient rule on
sinh x / cosh x:
Derivation
→ Quotient rule: d/dx(sinh x / cosh x) = [cosh x · cosh x − sinh x · sinh x] / cosh²x
→ Numerator: cosh²x − sinh²x = 1 (the hyperbolic Pythagorean identity from Ch.1)
→ So we're left with 1/cosh²x = sech²x
→ d/dx(tanh x) = sech²x
The other three — coth x, sech x, cosech x —
all come from writing them as reciprocals or quotients of sinh/cosh and applying
the quotient rule the same way. You won't usually be asked to reproduce these
derivations in an exam, but you are expected to know the results, so here's
the full table:
| f(x) | f′(x) | compare to trig |
|---|
| sinh x | cosh x | same pattern as sin → cos |
| cosh x | sinh x | ⚠ trig has cos → −sin, this one has no minus |
| tanh x | sech²x | same pattern as tan → sec² |
| coth x | −cosech²x | same pattern as cot → −cosec² |
| sech x | −sech x tanh x | ⚠ trig has sec → +sec tan, this one has an extra minus |
| cosech x | −cosech x coth x | same pattern as cosec → −cosec cot |
⚡ The one thing to actually memorise here
Four of the six hyperbolic derivatives are identical in sign to their trig
equivalents. Only cosh and sech break the pattern — cosh loses its
minus sign, and sech gains one it "shouldn't" have. If you can remember just
those two exceptions, you can reconstruct the whole table from your trig knowledge.
Combining with chain rule, product rule, quotient rule
What to Memorise
Exam Tips