Hyperbolic functions are built from exponential functions and appear in calculus, physics, engineering, and geometry. The main functions are sinh x, cosh x, tanh x, coth x, sech x, and csch x. Their graphs look related to ordinary trigonometric functions, but they describe hyperbolas, exponential growth and decay, hanging cables, and relativistic motion.
Knowing their derivatives helps you model change whenever exponentials combine in symmetric ways.
The derivatives of hyperbolic functions follow directly from their exponential definitions, such as sinh x = (e^x - e^-x)/2 and cosh x = (e^x + e^-x)/2. A key pattern is that the derivative of sinh x is cosh x, and the derivative of cosh x is sinh x. The functions tanh x, sech x, csch x, and coth x have derivative rules that often resemble trigonometric derivative rules but with important sign differences.
These rules are especially useful when solving differential equations, evaluating slopes, and simplifying expressions involving exponentials.
Understanding Calculus: Derivatives of Hyperbolic Functions
The derivative patterns come from one important detail about exponentials. When an exponent contains negative x, its derivative brings out a negative one. In the hyperbolic sine calculation, that new negative sign cancels the subtraction already present.
In the hyperbolic cosine calculation, it changes the addition into a subtraction. This explains why the two functions produce each other under differentiation. It is more useful to understand this sign change than to memorize a pair of isolated rules.
The identity hyperbolic cosine squared minus hyperbolic sine squared equals one gives another check on the pattern. Differentiate both sides using the product rule. The result simplifies to zero, as it must for a constant.
Hyperbolic tangent is a ratio of hyperbolic sine to hyperbolic cosine. Its derivative can be found with the quotient rule, then simplified by using the identity involving their squares. This simplification explains why its rate of change is always positive where it is defined.
Its graph rises steadily but levels off near positive one and negative one. Near zero, it behaves much like a straight line, so small inputs produce nearly equal outputs. Far from zero, much larger changes in input cause only small output changes.
This makes hyperbolic tangent useful for models with a natural upper and lower limit. Reciprocal hyperbolic functions need more care.
Hyperbolic cosecant and hyperbolic cotangent are not defined at zero, since their denominators become zero there. A derivative rule does not remove this restriction.
The chain rule matters whenever the input is more complicated than plain x. For example, the derivative of hyperbolic tangent of three t is three times the square of hyperbolic secant of three t. The three appears because the inner expression three t changes three times as fast as t.
If the input is t squared, the derivative includes two t as well. Product rules appear in expressions such as t times hyperbolic sine of t. First differentiate t while keeping the hyperbolic term unchanged.
Then keep t unchanged while differentiating the hyperbolic term. Students often lose a factor from the inner function or forget which part stays fixed during each product rule step.
These functions appear in physical situations where two opposite exponential effects are present. The shape of a uniform hanging cable is described by a hyperbolic cosine curve, called a catenary. Its slope involves hyperbolic sine, so calculus connects the cable shape to how steeply it rises away from its lowest point.
In differential equations, combinations of growing and decaying exponentials often become cleaner when written with hyperbolic functions. When studying them, pay close attention to signs. Ordinary trigonometric identities contain a plus in the familiar sine and cosine square relation, while the corresponding hyperbolic identity contains a minus.
That difference affects derivative simplifications, graph shapes, and solutions to equations. Checking values near zero, along with checking where a function is undefined, catches many mistakes.
Key Facts
- sinh x = (e^x - e^-x)/2 and cosh x = (e^x + e^-x)/2
- d/dx[sinh x] = cosh x
- d/dx[cosh x] = sinh x
- d/dx[tanh x] = sech^2 x
- d/dx[sech x] = -sech x tanh x and d/dx[csch x] = -csch x coth x
- d/dx[coth x] = -csch^2 x, for x not equal to 0
Vocabulary
- Hyperbolic function
- A function defined using combinations of e^x and e^-x that is related to the geometry of a hyperbola.
- sinh x
- The hyperbolic sine function defined by sinh x = (e^x - e^-x)/2.
- cosh x
- The hyperbolic cosine function defined by cosh x = (e^x + e^-x)/2.
- tanh x
- The hyperbolic tangent function defined by tanh x = sinh x / cosh x.
- sech x
- The hyperbolic secant function defined by sech x = 1 / cosh x.
Common Mistakes to Avoid
- Changing the sign in d/dx[cosh x]. The derivative is sinh x, not -sinh x, because differentiating e^-x creates a sign change that turns the plus in cosh x into the minus in sinh x.
- Treating hyperbolic derivatives exactly like trigonometric derivatives. Some rules look similar, but d/dx[cosh x] = sinh x while d/dx[cos x] = -sin x.
- Forgetting the chain rule with hyperbolic functions. For example, d/dx[sinh(3x)] = 3 cosh(3x), not just cosh(3x).
- Using coth x or csch x at x = 0. These functions are undefined at x = 0 because sinh 0 = 0 appears in the denominator.
Practice Questions
- 1 Find the derivative of f(x) = 4sinh x - 7cosh x + 2tanh x.
- 2 Find dy/dx for y = sech(5x) and evaluate the derivative at x = 0.
- 3 Explain why the derivative of cosh x is positive for x > 0 but negative for x < 0, using the graph of sinh x.