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

f(x) = √coth 3x

검증된 단계별 안내

Step 1: Recognize that the function f(x) = √coth(3x) involves a composition of functions. Specifically, it is the square root function applied to the hyperbolic cotangent function, which itself depends on 3x.

Step 2: Use the chain rule to differentiate the outer function √u, where u = coth(3x). The derivative of √u with respect to u is (1 / (2√u)).

Step 3: Differentiate the inner function coth(3x) with respect to x. Recall that the derivative of coth(u) is -csch²(u), where u = 3x. Then apply the chain rule to account for the derivative of 3x, which is 3.

Step 4: Combine the results from Steps 2 and 3. The derivative of f(x) is (1 / (2√coth(3x))) multiplied by the derivative of coth(3x), which is -3csch²(3x).

Step 5: Simplify the expression for the derivative. The final derivative is f'(x) = (-3csch²(3x)) / (2√coth(3x)).

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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 allows us to determine the slope of the tangent line to the curve of a function at any given point. The process of finding a derivative is called differentiation, and it involves applying specific rules and formulas, such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula used to compute the derivative of a composite function. If a function is composed of two or more functions, the chain rule states that the derivative of the outer function is multiplied by the derivative of the inner function. This is particularly useful when dealing with functions that involve nested expressions, such as f(g(x)).

Hyperbolic Functions

Hyperbolic functions, such as coth, sinh, and cosh, are analogs of the trigonometric functions but are based on hyperbolas instead of circles. The function coth(x) is defined as the ratio of the hyperbolic cosine to the hyperbolic sine, and it plays a significant role in calculus, especially in the context of derivatives and integrals involving exponential functions. Understanding their properties is essential for differentiating functions that include hyperbolic terms.

22–36. Derivatives Find the derivatives of the following functions.f(x) = √coth 3x