Differentiating hyperbolic functions

Hyperbolic functions, introduced in Pure Mathematics 1, are a key topic in Core Pure Mathematics 2. They are frequently used in calculus for their unique differentiation and integration properties.
The two primary hyperbolic functions to initially focus on are sinh x and cosh x, which are defined respectively as ((e^x) - (e^-x))/2 and ((e^x) + (e^-x))/2.
The derivative of sinh x is cosh x. This is a crucial result to know as it stands at the core of differentiating hyperbolic functions. This means that if y = sinh x, then dy/dx = cosh x.
Likewise, the derivative of cosh x is sinh x, so if y = cosh x, then dy/dx = sinh x. Note the similarity these have to the derivatives of sin x and cos x; however, there are no negatives involved with hyperbolic functions.
This pattern extends to hyperbolic functions of higher powers. The derivative of (sinh x)^n or (cosh x)^n can be found using the chain rule, with the end result resembling the derivative of a sinusoidal function raised to the nth power.
Hyperbolic functions also obey the product and quotient rules. If you are differentiating a product or quotient involving hyperbolic functions, apply these rules as usual.
Last but not least, the chain rule also applies when differentiating composite hyperbolic functions. For instance, if you’re asked to differentiate sinh(2x), substitute u = 2x, then apply the chain rule: du/dx * d/dx (sinh u).
One of the trigonometric identities that can be used for differentiating more complex hyperbolic functions is the double-angle formula: cosh(2x) = cosh^2(x) + sinh^2(x).
Unlike circular functions, hyperbolic functions are not periodic, and their derivatives do not follow the same repeating pattern. They do, however, reflect the same symmetry properties, where the derivative of sinh is cosh, and vice versa.
The main takeaway is this: differentiation rules that apply for standard functions equally apply for hyperbolic functions. That includes product and quotient rules, chain rule, power rule, and more. With these in mind, differentiating hyperbolic functions can be a straightforward process.