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

f(x) = sinh 4x

검증된 단계별 안내

Step 1: Recall the derivative rule for hyperbolic sine functions. The derivative of sinh(u) with respect to x is cosh(u) * du/dx.

Step 2: Identify the inner function u in the given function f(x) = sinh(4x). Here, u = 4x.

Step 3: Compute the derivative of the inner function u = 4x with respect to x. The derivative of 4x is 4.

Step 4: Apply the chain rule. Substitute u = 4x into the derivative formula: f'(x) = cosh(4x) * 4.

Step 5: Simplify the expression to write the derivative as f'(x) = 4 * cosh(4x).

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주요 개념

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Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the function's graph at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule.

Hyperbolic Functions

Hyperbolic functions, such as sinh (hyperbolic sine) and cosh (hyperbolic cosine), are analogs of the trigonometric functions but are based on hyperbolas instead of circles. The function sinh(x) is defined as (e^x - e^(-x))/2. Understanding these functions is crucial for differentiating expressions involving them, as they have specific derivatives that differ from their trigonometric counterparts.

Chain Rule

The chain rule is a fundamental technique for finding the derivative of 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 differentiating functions like f(x) = sinh(4x), where the inner function is 4x.

22–36. Derivatives Find the derivatives of the following functions.f(x) = sinh 4x