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.
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.