13–40. Evaluate the derivative of the following functions.
f(x) = 1/tan^−1(x²+4)

검증된 단계별 안내

Step 1: Recognize that the function f(x) = \(\frac{1}{\tan^{-1}\)(x^2 + 4)} is a composition of functions, where the outer function is g(u) = \(\frac{1}{u}\) and the inner function is u(x) = \(\tan\)^{-1}(x^2 + 4).

Step 2: Apply the chain rule for differentiation, which states that the derivative of a composite function g(u(x)) is g'(u(x)) * u'(x).

Step 3: Differentiate the outer function g(u) = \(\frac{1}{u}\) with respect to u to get g'(u) = -\(\frac{1}{u^2}\).

Step 4: Differentiate the inner function u(x) = \(\tan\)^{-1}(x^2 + 4) with respect to x. Use the derivative formula for \(\tan\)^{-1}(v), which is \(\frac{1}{1+v^2}\), and apply it to v = x^2 + 4. Also, apply the chain rule to differentiate x^2 + 4.

Step 5: Combine the results from Steps 3 and 4 using the chain rule: f'(x) = g'(u(x)) * u'(x). Substitute back u(x) = \(\tan\)^{-1}(x^2 + 4) and simplify the expression.

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주요 개념

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Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function's graph at any given point. The derivative is often denoted as f'(x) or df/dx and can be calculated using various rules such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(u), where u = g(x), then the derivative of the composite function f(g(x)) is f'(g(x)) * g'(x). This rule is essential when differentiating functions that involve nested expressions, such as the function in the given question.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan^−1(x), are the functions that reverse the action of the standard trigonometric functions. For example, tan^−1(x) gives the angle whose tangent is x. When differentiating functions involving inverse trigonometric functions, specific derivative formulas apply, such as the derivative of tan^−1(x) being 1/(1+x²), which is crucial for evaluating the derivative of the given function.

13–40. Evaluate the derivative of the following functions.f(x) = 1/tan^−1(x²+4)