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Differentiating hyperbolic functions

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

The derivatives of the hyperbolic functions follow straight from the exponential definitions. They look like the trigonometric results, but watch the signs: d/dx(cosh x) is +sinh x.

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The results

Differentiate the definitions: d/dx (e^x - e^-x)/2 = (e^x + e^-x)/2, so d/dx(sinh x) = cosh x.

y	dy/dx
sinh x	cosh x
cosh x	sinh x
tanh x	sech² x
sech x	-sech x tanh x
cosech x	-cosech x coth x
coth x	-cosech² x

sinh, cosh and tanh are in the formula booklet; the reciprocal results can be derived with the quotient or chain rule.

Proving a result

For tanh x use the quotient rule: d/dx(sinh x/cosh x) = (cosh² x - sinh² x)/cosh² x = 1/cosh² x = sech² x.

For sech x write (cosh x)^-1: d/dx = -(cosh x)^-2 sinh x = -sech x tanh x.

Combining with the chain, product and quotient rules

d/dx cosh(3x) = 3 sinh(3x)
d/dx (x sinh x) = sinh x + x cosh x
d/dx ln(cosh x) = sinh x/cosh x = tanh x

Stationary points: solve dy/dx = 0, which often gives tanh x = k and x = artanh k = (1/2) ln((1 + k)/(1 - k)).

Worked example

Find the stationary point of y = 5 cosh x - 3 sinh x.

dy/dx = 5 sinh x - 3 cosh x = 0
tanh x = 3/5, so x = (1/2) ln((1 + 3/5)/(1 - 3/5)) = (1/2) ln 4 = ln 2
cosh(ln 2) = 5/4 and sinh(ln 2) = 3/4, so y = 25/4 - 9/4 = 4

Answer: (ln 2, 4), a minimum

Key idea: d/dx sinh x = cosh x; d/dx cosh x = sinh x (no minus); d/dx tanh x = sech² x. Use the chain, product and quotient rules as usual.

Check yourself

y = cosh 3x. Find dy/dx when x = 0.2, to 3 decimal places.
Show answer
1.910. dy/dx = 3 sinh 3x = 3 sinh 0.6 = 1.910.
What is d/dx(cosh x)?
- -sinh x
- cosh x
- sech² x
- sinh x
Show answer
sinh x. d/dx (e^x + e^-x)/2 = (e^x - e^-x)/2 = sinh x.
y = x sinh x. Find dy/dx when x = 1, to 3 decimal places.
Show answer
2.718. dy/dx = sinh x + x cosh x. At x = 1: sinh 1 + cosh 1 = e = 2.718.
Find the x-coordinate of the stationary point of y = 5 cosh x - 3 sinh x, to 3 decimal places.
Show answer
0.693. 5 sinh x = 3 cosh x, so tanh x = 3/5 and x = ln 2 = 0.693.
Find the minimum value of y = 5 cosh x - 3 sinh x.
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4. At x = ln 2: cosh = 5/4 and sinh = 3/4, so y = 25/4 - 9/4 = 4.
What is d/dx ln(cosh x)?
- coth x
- 1/cosh x
- tanh x
- sinh x
Show answer
tanh x. Chain rule: (1/cosh x) × sinh x = tanh x.
y = sech x. Find dy/dx when x = 1, to 3 decimal places.
Show answer
-0.494. dy/dx = -sech x tanh x = -0.648 × 0.762 = -0.494.

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Sources

Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

Last updated 8 November 2026. Spotted a mistake? Tell us.