22–36. Derivatives Find the derivatives of the following functions.


f(x) = cosh²x

Verified step by step guidance

1. Step 1: Recognize that the function f(x) = cosh²(x) is a composite function. It involves the square of the hyperbolic cosine function, cosh(x). To differentiate it, we will use the chain rule.
2. Step 2: Recall the chain rule formula: If a function is composed as g(h(x)), then its derivative is g'(h(x)) * h'(x). Here, g(x) = x² and h(x) = cosh(x).
3. Step 3: Differentiate the outer function g(x) = x². The derivative of x² is 2x. Applying this to g(h(x)), we get 2 * cosh(x).
4. Step 4: Differentiate the inner function h(x) = cosh(x). Recall that the derivative of cosh(x) is sinh(x).
5. Step 5: Combine the results from Step 3 and Step 4 using the chain rule. The derivative of f(x) = cosh²(x) is 2 * cosh(x) * sinh(x).

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a specific point.

Hyperbolic Functions

Hyperbolic functions, such as cosh(x), are analogs of trigonometric functions but are based on hyperbolas instead of circles. The function cosh(x) is defined as (e^x + e^(-x))/2 and is used frequently in calculus, particularly in problems involving exponential growth and decay. Understanding their properties is essential for differentiating functions that involve them.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions like cosh²(x), where one function is nested within another.